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.
Lifetime Sales Predictions
Switch: 151 million (was 73, then 96, then 113 million, then 125 million, then 144 million)
PS5: 115 million (was 105 million) Xbox Series S/X: 57 million (was 60 million, then 67 million)
PS4: 120 mil (was 100 then 130 million, then 122 million) Xbox One: 51 mil (was 50 then 55 mil)
3DS: 75.5 mil (was 73, then 77 million)
"Let go your earthly tether, enter the void, empty and become wind." - Guru Laghima