To belabor my point about Duverger's law and why it doesn't mean third parties shouldn't run at all.

In 168 House districts Trump got only 40% of the vote or less in 2016 -- according to The Cook Political Report . If you want a more cushy threshold we can say in 92 House districts Trump got only 30% of the vote in 2016 or less. If we include all of California's House districts because they have a jungle primary where the top two candidates move on to the general election regardless of party status, that means the People's Party (or any other left-wing third party) can run in somewhere between 100 to 200 House districts (again depending on which threshold you prefer) without jeopardizing the district to far-right Republicans and either the PP or Dems would win an overwhelming majority of those seats. 

The result would be that it would be a competition between Democrats and PP candidates in these districts, and since the PP candidates would be challenging Democrats to the left they'd pull their Democratic competitors to the left. Republicans voters in those districts would have to decide whether they'd want to vote Republican or not because chances are that the GOP would be a third party in this situation. So they might cross over to vote Dem instead of Rep so that the PP doesn't win, if they prefer Dems to PP. 

Assuming the People's Party only gets a third of the seats, Democrats get 65% of them, and moderate Republicans get 2%, that'd give PP somewhere between 33-66 seats (depending on threshold), Dems somewhere between 65 - 130 seats (depending on threshold), and moderate Republicans somewhere between 2-4 seats (depending on threshold.) This might sound bad to give Republicans 2-4 seats they'd otherwise not have, but the ideological median likely would be further left than it would've been otherwise because the PP would pull Democrats leftward, and Democrats would pull moderate Republicans leftward in these districts. 

It makes the chances of progressive legislation passing much higher as the whole Overton Window has shifted leftward when you have a House of Representatives with say 66 Populists (synonym for People's Party members), 162 Democrats, 1 Libertarian, and 202 Republicans with the Populists and Democrats forming voting coalitions for a majority and to elect the Speaker of the House, but otherwise having separate infrastructures. 

Here is a map of Canada's 2015 election. Canada has a FPTP system, just like the United States. You'll notice that four parties had representation 

Here is a 2019 electoral map. 

Notice that five parties have representation. 

Here is a map of the 2015 U.K elections, like Canada and the U.S they have a FPTP system. 

Here is a map of the 2019 U.K elections. 

Duverger's law tells us that there will be two major parties in any system based on FPTP, but it doesn't say that third parties will have absolutely no influence or ability to influence elections. 

Like I noted in my previous post, roughly 90 House seats are solidly Democratic to the point that only 30% (or less) of their constituents vote Republican. This means that the vote could split down the middle Democratic/left-wing third party 35% for each, and one of them would still win because the Republican only gets 30% (at max) of the vote normally. 

Duverger's law doesn't apply in the situation where one of the national parties are so weak in a certain region to the extent that it becomes essentially a third party in that region. 

This is why you find Lib Dem/Conservative constituencies in Britain, or NDP/Conservative ridings in Canada. There is no reason why there can't be Populist Party/Democratic districts or Populist Party/Republican Party districts in the U.S where those are the two major parties of that district. 

And there lies the hen-and-egg problem.

Neither Democrats or Republicans would ever change the system unless forced to, as it would mean they'd loose power to other parties. In other words, you can kiss goodbye the dream of winner-takes-all go away anytime soon - or ever, really.

Since the states choose themselves what system they use to vote in the presidential election, they need to win states (I think governor, or does it need a different post to be able to change state voting laws?) to be able to slowly change the voting laws in the states one by one until the number of states with winner-takes-all is relatively small and don't singlehandedly decide over the outcome in elections. Only then can third parties really attack the presidency.

The problem: They need to run in the next presidency to make themselves both visible to people who don't follow politics closely and for their own credibility. But that doesn't mean they'd have to run in swing states yet to not hand the election to republicans on a silver platter. Or that they even get widespread ballot access in the first place for that matter.

Fortunately for many states all that is needed to change the voting system (likely to ranked choice, but maybe proportional or a mixed system in a few states as well) is a ballot initiative. This is how Maine got ranked-choice voting a few years ago. 

Here are the states that allow ballot initiatives from their populations. If the population votes yes, then the legislature and governor have no choice but to enact their vote.  

Then the remaining states that don't have initiatives are going to have to get a coalition of third parties to push for either the major parties to change the system, or otherwise they'll run a legislative reform campaign designed to oust the major parties and get that sole reform done. 

Also See: 

Well if people don't think they are represented by any of the 2 parties then it is good that they try to make a new one. But they should have to look at Brazil and the several dozen parties here. It isn't much better to have plenty of parties versus having 2 parties with several sub parties inside.

There is one major difference between the FPTP in Canada and the UK compared to the US: They're local elections while in the US, it's almost always entire states. I would agree with your assessment if the US had all their states divided up into voting districts like in Maine for instance, but the way it is, entire states are just too big in weight for most voters to go off the GOP/Dem rails.

But like you said in the other post, if voters can make the choices themselves on how they would like to vote, then getting more equitable voting systems in place should be a real possibility, at least in those states that allow those.

House of Representative elections are not state-wide elections, nor are state-senators and state House of Representative/assembly elections, nor are local elections. 

Sure the Senate and Presidency are elected state-wide or through a system of fifty one (state + D.C) elections, but they are only two parts of the whole system.

Third parties and independents have won Senate seats in the past too, and two Senate seats are held by independents currently (Vermont - Bernie Sanders, and Maine - Angus King.) 

The most successful third party in history, by far, was the American Independent Party. The entire basis of the AIP's existence was white Southern rage over the Civil Rights Act of 1964, which was passed entirely along geographic lines. The opposition was almost every Southern legislator, regardless of political party, with Southern Democrats and Southern Republicans alike voting against it, plus a handful of non-southern Republicans. All but two or three Democratic legislators outside of the south voted to pass the Civil Rights Act. The only non-southern Democrat in the Senate to vote "nay" was Robert Byrd from the border state of West Virginia, who filibustered the Act for 14 hours. President Lyndon Baines Johnson, a Texan and a Democrat, worked at getting as many Southern Democrats as he could to cross the aisle and support the Civil Rights Act, with some success, and he signed the bill into law despite Southern opposition and anger. 

One of the "nay" votes came from Senator Barry Goldwater of Arizona. Goldwater was the Republican candidate for president in 1964. Johnson won the 1964 Presidential election in one of the largest electoral landslides in American history, and with the largest popular vote margin in American history. Goldwater won only six states, but the pattern of states he won attracted a lot of interest from GOP leadership. All of the five states that Goldwater won, other than his home state of Arizona, were Deep South states that up until then had been part of the Democratic "Solid South" - Louisiana, Mississippi, Alabama, Georgia, and South Carolina, angered over the passage of the Civil Rights Act. They voted for an Arizona Republican over a Southern Democrat. Goldwater's reason for voting against the Civil Rights Act was that he felt it was a violation of states' rights, which was the same argument that was used to oppose abolition and desegregation. The other Senator from Arizona, Democrat Carl Hayden, voted to pass the Civil Rights Act.

The Republican Party capitalized on Goldwater's vote to appeal to Southern voters, but they ended up losing the rest of the country in the process that year because most of the country outside of the South was alienated by the idea of open segregation, even though there was plenty of de facto segregation outside of the South thanks to redlining, which was the practice of keeping blacks out of white suburbs by refusing to sell to them and denying them mortgages to buy homes in the suburbs. Segregation was apparently okay as long as it wasn't called segregation. 

The American Independent Party was an extreme right-wing party founded on open support of segregation and "law and order," which was coded political language for keeping minorities out of the white suburbs that sprung up as a result of white flight. In 1968, they nominated George Wallace, the Democratic governor of Alabama famous for trying to block black students from entering an all white school with his own body and his inauguration speech as governor, "Segregation now, segregation tomorrow, segregation forever".  Wallace didn't run as a Democrat because the mainstream Democratic Party by that point had completely rejected overt segregation (meanwhile, blacks in non-southern states continued to be redlined). He said that "there isn't a dime's worth of difference between the two parties." Wallace's supporters included segregationists, radical right-wing supporters in California, and the John Birch Society. 

Almost as controversial as Wallace himself was his running mate, Curtis LeMay, the former Chief of Staff of the USAF.  LeMay advocated using nuclear weapons in Vietnam and said that the public had a "phobia" of nuclear weapons. 

Wallace ran on segregation and "law and order," which was coded language (dog-whistling) for keeping minorities out of white suburbs and confined to the inner cities. Nixon adopted a "law and order" campaign in response, wanting to appeal to Southern voters over racial angst while also not wanting to alienate Northern and Midwestern conservatives by openly advocating for segregation. The Democratic candidate, Hubert Humphrey of Minnesota, also felt forced to run on a "law and order" campaign so as to not appear soft on crime and violence. The violent "long, hot summer" of 1967 and the continued unrest in 1968 forced his hand on the issue.

The Democratic Party was in disarray. Johnson was initially running for reelection, but by that time, the public strongly disapproved of his handling of the Vietnam War, and the Tet Offensive was the last nail in Johnson's presidential coffin. Johnson was also in poor health and ultimately dropped out. Robert F. Kennedy had the support of a broad coalition of people who opposed the Vietnam War, Blacks, Catholics, and poor whites, and was heavily favored to win the election in 1968, but he was assassinated in June. Hubert Humphrey was the eventual nominee, but was unpopular because he wouldn't commit to ending the war, which aroused the ire of antiwar activists and led to the 1968 DNC Riots in Chicago. The Democrats were splintered while the Republicans were united, and that put Nixon in office. 

Wallace won 10 million votes and five Southern states, plus one faithless elector in North Carolina, which otherwise voted for Nixon. Nixon only got a half-million more votes than Humphrey but still won with a huge electoral majority. Wallace was by far the most successful third party candidate in the history of the United States, all on a platform of racial hatred. The American Independent Party faded out after the 1968 election. 

In 1972, Wallace ran as a Democrat. Segregation had turned into a third rail, so he renounced his former segregationist stance. He was shot during a political campaign in Maryland. He survived, but it put an end to his presidential campaign. He was a paraplegic for the rest of his life. Nixon won 49 states in 1972 and swept the entire South for the first time in the country's history. George McGovern, the Democratic candidate, only won Massachusetts, the liberal home state of John and Robert Kennedy.  

But the Republicans had seen the pattern of states that Goldwater and Wallace had won, they saw an opportunity to bring new blood into the party, and they took it. Some Southern Democrats, including Wallace, repented of their former segregationist stance. Wallace said in 1979, "I was wrong. Those days are over, and they ought to be over." Byrd, the WV senator who had filibustered the Civil Rights Act, also renounced his former views and expressed shame over his own history in Congress. But Strom Thurmond of South Carolina became a Republican and helped bring more Southern Dems into the Republican tent, as did Jesse Helms of North Carolina, who was a Republican for his entire Senate career.  Since segregation was now a third rail, they instead made those same promises of "law and order" to white flight suburban voters. The GOP also began infusing evangelical Christianity into the party's DNA. 

But Wallace had dragged the entire Republican Party to the right, and since the Democrats didn't want to appear to be soft on crime, they too had to advocate for "law and order."  So the Democrats also got dragged to the right. 

I hope all of this illustrates some of the problems with third party voting in the United States. This was by far the most successful third party in American history, and it ran on a platform of racial hatred, with a lunatic who though the casual use of nuclear weapons was okay thrown in for good measure. Republicans today are smart enough not to be distracted by third parties. They're keeping their eyes on the prize. 

Sure but that doesn't matter in the US system. One party could technically own 100% of both House and Senate yet not win the presidential election. To make that matter, the presidential election would need to stop being a separate thing from either House of Senate election.

In fact, this could actually be done very easily: There are 538 electoral votes in total. There are 438 Representatives in the House plus 100 Senators in Congress. Just add up both houses by party affiliation and voila: A presidency elected by the result of many local elections, where smaller and regional parties actually have a fighting chance, ending the deadlock between just 2 parties.

A few things to note: 

1. Canada has universal healthcare in a large part because the CCF (the direct predecessor to the NDP) existed. Read about Tommy Douglas. Your hyperfocus on whether or not the parties get a majority, rather than whether or not the parties represent majority positions is skewed, in my opinion. I can guarantee you that despite most of the elections ending in a mere plurality result, Canadians are better represented in their parliament than Americans are in Congress, and that is because Canadians have more competitive elections. 

2. You don't play a spoiler when you're the second or first most popular party. You can't argue that if the vote is 45% Conservative, 30% NDP, and 25% Liberal that it is the NDP playing spoiler. 

3. The NDP has been very, very, very, very successful at getting the Liberal party to move left. They had roughly the same policies in 2015 because the NDP was the second largest party in 2011. See: https://en.wikipedia.org/wiki/2011_Canadian_federal_election

4. Regional parties have existed in the U.S too. The Northern vs. Southern Democrats are an obvious example, but you also had examples like the Minnesota Farmer-Labor Party, which is actually the second party in that state today, yes they were so successful that they replaced the Democrats and now form a coalition with the national democrats in the form of the Minnesota Democratic-Farmer-Labor Party. 

Presidential elections are not national elections, they are 51 separate state + D.C elections, but it really doesn't matter anyway. A successful third party doesn't need to run a successful presidential campaign to succeed in its goals. There is much more to politics in the United States than presidential politics. State laws, local laws, and who is in Congress matter quite a bit too. There is no reason why there couldn't be coalitions in the U.S congress. The Speaker of the House could easily be voted on by a majority coalition of say 166 Democrats and 66 Populists or whatever other combination.