Let's look at the Xbox and PlayStation.

The 360 had a bunch of top-selling exclusives, and 6 of its Top 20 selling games are exclusive games. The X1 only has 4 exclusives in its Top 20, and they didn't sell as well in comparison. That, as much as the PlayStation's successes, is why the X1 is doing worse than the 360 did.

The PS3 and PS4 have a similar number of high-selling exclusives, but the PS4 had the advantage of getting these games within the first three years of its life. The PS3's big exclusives besides MGS4 didn't come until 2009 or later, while the PS4 had some solid exclusives by mid-2016 (Uncharted 4, Uncharted Collection, Last of Us Remastered, Killzone: Shadow Fall, and Ratchet & Clank). Lo and behold, the PS4 had better early years than the PS3 (though this clearly wasn't the only factor).

They are quite important.

For Nintendo, they are do or die. Even Nintendo platforms with large third-party support (either by numbers or quality), still are moved mostly by first-party games. The NES/Famicom could still have the third-party library it has, but without Super Mario Bros., The Legend of Zelda, and Duck Hunt it wouldn't go far.
They helped Sega a lot as well. Now I get that Sega's only sure-fire hardware hit was the Mega Drive/Genesis, but Sega's games attributed a lot to that platform's success.
Microsoft is interesting. Halo: Combat Evolved single handedly kept the Xbox console from failing. I say that because the hardware sales of the Xbox were nothing to celebrate. But Halo, Halo 2, and Xbox Live ensured that a future Xbox was possible. Xbox 360 thrived with first-party games, but a lot with third-parties as well. And the Xbox One has very few true exclusives. And the first-party games it has aren't very acclaimed.
Sony seems to be the only one who can coast by without massive first-party support. Don't get me wrong, first-party games have helped make PlayStation what it is. But it wasn't Gran Turismo (though this game helped), Spyro, or Ape Escape that turned the tide to make the PS1 a titan. It was Final Fantasy VII. And it was GTA (a timed third-party exclusive) that helped contribute to the massive success of the PS2. PS1 and PS2 had massive amounts of exclusives that weren't first-party. With PS3 and PS4, that has changed due to increased competition from Nintendo and Microsoft, as well as PC.

Take away first party from all the big three, how well would the 3 current consoles be selling? Sony and Microsoft will be doing just fine but Nintendo, who relies heavily on their first party games, won’t do well as their third party support hasn’t been great over previous gens.

Now, put back in first party and take third party away, Nintendo and Sony will do just fine, but Microsoft, wouldn’t do as well as their first party support hasn’t been great this gen compared to previous gens.

It’s like Sony is a balanced seesaw of first and third party support while the other two leans all the way on each end. Sony’s mastered the art of gaming for everyone.

Hopefully the series x and Microsoft’s acquisitions helps boost Xbox’s first party support next gen.
Xbox needs games that make people go, oooh I’m interested in this game, I’ll get an Xbox, not oooh, I’m interested in this game, I’ll get it on PC even though it’s also on Xbox. (it’s a win win for Microsoft but a win lose for Xbox)
I believe that Xbox could be around 60-65m if their games stayed full exclusive or released on pc a couple years after each game launch.

Xbox sales may not have been massive, but for their first system and outselling GC with a great help from Halo is really something.

The games aren't running on PCs though. They're running on Google's servers at their Data Centers. Stadia is a platform, it's not a store like Epic Games is. It's Google's answer to PlayStation, Xbox, and Nintendo Switch.

Sony was never really a slouch in the first party department though. Crash, Spyro, GT, Parappa and Twisted Metal all helped define the PlayStation as much as Tomb Raider or FFVII. Even with the PS2, many of the most critically acclaimed and best selling games on the system were published by Sony Computer Entertainment. 

But nobody's using Stadia, that's the problem. Even after the service became free to use, nobody's buying anything on it. That's why it needs first party games, something you can only get on Stadia.

Stadia isn't a store, it's a platform because its running on its own hardware with its own services and features. 

Steam is a service where you download PC games and run them on your own PC Hardware. Stadia isn't like that. The games are running on Google's own hardware, and streamed to the user like a YouTube video. It's very different. 

Sony would struggle to sell 25 million units if you took their third party games away, don't kid yourself. No COD, no GTA, no Final Fantasy, no NBA 2K, no FIFA, no Madden, no Red Dead = game over. 

The only similarities is that its free to use and let's you use different controllers. Otherwise, it's more akin to PS and Xbox than Steam. Steam is also technically a platform. But it requires a PC to download and run games. Stadia let's you use any tablet, phone, or Chromecast to play games.