Nintendo's strategy has always been for first party to do the heavily lifting during the first couple of years, and then third parties sell the system for the rest of the system's life.  If you look at every successful Nintendo system this is exactly how it went.  If you look at every system that flopped, it's because the third parties never came.  I'm not even talking AAA third parties.  It's more about quantity than quality.  The term AAA doesn't even make sense on handheld systems, but Nintendo handhelds have always had a lot of third party games, and they always sell well.  The real question is, "Does the Nintendo system have a lot of third party games?"  If "yes", then it's successful.

The clearest example of how third party games impact Nintendo is the N64.  It started out fairly strong, but then sales fell quickly.  It just didn't have the third party games to keep it going.  The N64 is what it looks like when Nintendo tries to sell a system on first party games alone.  Switch has a lot of third party games and every year it gets more than the previous one.  This is why even when hardware sales start to decline, it will be a fairly gradual decline.  It's the third party games that keep a system from dropping fast.  

Here they are. The data is directly taken from this site's sales numbers.

And for better comparison, I also added the numbers for both PS4 and Xb1

I only picked the best selling game per franchise, because having 3, 4 or more of the same "game" distorts the picture quite heavily.

I also ignored GTA V for obvious reasons.

I think so, but even a mixing of 1st and 3rd party support isn't the biggest reason by themselves

The biggest reason imo is the fact it's hybrid; You can play AAA games in your bed, and in a train, in a queue, it's awesome

But of course, it would be meaningless if you don't have such games, which happens to be both 1st and 3rd party

So I'd say there isn't only one big factor to this, each small "factor" is feeding other another. If you remove a key aspect of Switch success everything would fall apart

On out list of best-selling Switch games, the Top 19 games are all Nintendo productions. So I'm tempted to say no.

That said, we have a handful of third party games with over a million copies confirmed (Mario + Rabbids, Minecraft, Octopath, Skyrim, FIFA, Taiko no Tatsukin), not to mention however many games that we don't have confirmed but can speculate have sold that well (Bomberman, Just Dance, Sonic Forces, Fortnite, etc).

Let's try another indicator: online retailers (at least in the USA). Of the Top 20 skus for Switch games on Gamestop, 5 are third party. From Best Buy, it's also 5 of the Top 20. From Walmart it's 9 of the Top 20, and from Amazon, it's 8 out of the Top 20. Notably, these games are not all the same five titles, though Minecraft is a recurring presence.

I guess overall we can give third parties roughly as much credit on the Switch as we can for the success of the Wii. It's a Nintendo-led success with 3rd party contiributions.

Third parties primarily interest those who already have a Switch. My guess would be that the majority of hardware sales come because people want to play Breath of the Wild, Mario Kart 8, and Animal Crossing. I think almost everyone I know purchased a Switch for Breath of the Wild or Mario Kart, and Animal Crossing caused a huge surge of sales. Although, looking at logues of people, almost everyone I know with a Switch plays more third party games than first (minus Mario Kart). I'm playing mostly RPGs and creative non-action adventure games, so aside from Xenoblade, Animal Crossing, and Mario Kart, my list is filled with third party games.

There are some are certain third party games that do really help sell a system like GTA 5, Madden, Monster Hunter World and Call of Duty but for the most part first party is the king.

The Switch has got some great third party support.
The big third party companies are the only ones that I feel like are lagging behind.
Of course you have Warner Bros who have brought all but three of their releases to the Switch since it launched in 2017 but then you have Bandai Namco & EA massively lagging behind. I suppose we'll see how it changes, especially with Bandai Namco games selling incredibly well on the Switch, they are sure to get things released at a better rate by next year.

Regular Switch is sold out everywhere here in the states. Switch Lite is most often in stock. I've no doubt that Switch Lite is pushing hard sales in Japan, but definitely not in NA, and probably not in Europe or Australia.

For those wondering, Nintendo said in their last financial report that 9 third party games sold at least 1 million copies during the 2020 FY. However, they didn't name what those games were. This is in addition to third party games that sold at least 1 million in previous years.

The bolded are actually first party.

It helps a lot, but I don't think it is why its selling so well... Marketing and stellar first party lineup are the most important factors I think.

I have to say though, I have never bought more third party games for any other system.