And what has the results been? Outcomes that are unrepresentative, sometimes incredibly so. The actual composition of their Houses of Commons is often way out of line with how people actually voted. Many MPs in recent elections won with less than 40% of the popular vote in their electoral district. As an example of these unrepresentative results, in the 2015 Canadian federal election, the Liberals won an absolute majority of seats with less than 40% of the national popular vote, while in 2019 they came in second and still won the most seats (though not a majority). The NDP had 16% of the vote but only 7% of the seats. In a good number of districts won by the Conservatives, the NDP appears to have served as little more than a spoiler. How many seats would the Liberals and Conservatives have won if it was just the two of them? The Conservatives don't have a rival right-wing party like the Liberals have three rival left-wing parties, and they were able to control the House of Commons for nearly a decade under Stephen Harper while never gaining more than 40% of the national vote.
Additionally, some of those smaller parties in Canada & the UK, like Bloc Quebecois, SNP, and Plaid Cymru, are limited to specific cultural regions that vote overwhelmingly for that party. In the UK, for example, England by itself is closer to a two-party system, with the Conservatives and Labour winning a combined 81.2% of the popular vote and nearly all the seats in Parliament in the last election, with the Lib Dems, Greens, and Brexit Party being incredibly underrepresented in proportion to their vote share, and probably also serving as spoilers in most elections. Aside from Labour, the Conservatives, and the Lib Dems, most of the other parties that hold seats in the British Parliament get the entirety of their representation in Parliament from Scotland, Wales, and Northern Ireland, and are, at present, generally over-represented.
Elections in the UK and Canada are a horrible mess because of the existence of more than two major parties. They make our elections look neat and tidy by comparison, and ours are already unrepresentative enough. FPTP rules are not the norm for lower house elections in most of the world, and it's easy to see why.
Also, those nations have a parliamentary system, while the United States has a full presidential system. When we talk about elections in the UK or Canada, we're talking purely about those lower house elections. The executive in parliamentary systems is an unelected but largely ceremonial office, while in the United States the president is elected by the people (albeit indirectly), exists separate from and co-equal to the Congress, and exercises actual power as, unlike in parliamentary systems, full separation of powers is unique to full presidential systems. The upper house is likewise unelected but with less power than the lower house in parliamentary systems, whereas our Senate exists co-equal to the House and its members have been directly elected since 1913. Their legislative process and general rules are different. There is no such thing as a minority government, hung parliament, or snap elections in the U.S. Either a majority of the members of Congress vote in favor of a bill or they do not. If they can't get stuff done at all, well, that may or may not get fixed next election, which will happen on a fixed date every even-numbered year and not a day before or after.
And when it comes to presidential elections, it's not a lot of smaller district-level elections. It's the whole country voting for one person. Whoever wins the most votes in a state wins that state's electoral votes. If the Democratic base was heavily split between two parties, most swing states would effectively become solid blue states. Republicans could win Florida, Ohio, Wisconsin, Pennsylvania, Michigan, and so on. Some states that are more firmly blue could reverse course and end up flipping to the red column as Republican candidates still manage to get 40-45% of the vote. All they would need to win is for that remaining 55-60% to be divided heavily enough. Even if we abolished the Electoral College and replaced it with a direct popular vote, the GOP candidate could still win the election with only 40-45% of the vote if the progressive vote is sufficiently split, and if they get above 45% but not quite 50%, they're almost guaranteed election if a major left-wing third-party consistently manages to nab at least 5-10% of the vote.
Realistically, though this proposed new party would itself never stand a chance of winning the presidency itself, given the ideological makeup of America. While self-identified "liberals" have been growing as a percentage of the population, they are still a minority, with self-identified moderates being a plurality. Not all of them are going to abandon the Democrats, either. Given the typical third-party vote on the left, I would imagine that the Democrats might lose at most about 10-15% of the progressive vote, or approximately 5-7% of their total base. That would never give this new party a chance at capturing the White House for a very long time, though it would be sufficient to cost the Democrats close races. In other words, they'd simply do what third parties have always done in America: serve as a spoiler.
The United States is not Canada. It is not the United Kingdom. We do not operate on the Westminster System. And in our system, third parties have only ever served as a spoiler. But in all of these countries, we need ranked-choice voting and, for lower house elections, proportional representation.