Nintendo made several eastern third parties pretty happy with the Switch, it seems. Just look at it:

Konami - Super Bomberman R - 500k sales. That's quite a lot!

Sadly, nothing else announced from them so far. But considering it's Konami, that's not a surprise.

Capcom - USFII and MHXX so far. Both will do pretty good. Of course MHXX will sell much more, but Street Fighter's sales will be good nontheless, relatively speaking to the resources put into it.

Sega - Puyo Puyo x Tetris sees strong sales so far. I hope Sonic Mania strikes hard as it fits right into the nostalgia train.

Square Enix - Dragon Quest XI and Dragon Quest Heroes I+II are a strong commitment. Those games will strike in Japan for sure. Project Octopath Traveler still has me excited. I hope it sells well.

Koei Tecmo - Fire Emblem Warriors should do quite okay if Hyrule Warriors can be used as an indicator.

I hope Nobunaga's Ambition comes to the west. Btw, they already announced a new one for next year, apparently

Atlus - Here's hope that Shin Megami Tensei becomes a hit.

Atlus is the developer, but it gets published by Sega

Nippon Ichi Software - Disgaea 5 saw high preorders of around 200k. That's a good number.

Now the only one left for me is Namco Bandai. They announced a Classics Collection in a Direct recently, but I honestly can't tell how good this will sell.

You're forgetting Dragonball Xenoverse 2; a new Tales and Taiko Drum have also been announced already, just like a deluxe edition of One Piece Unlimited World Red

Western third parties don't give a fuck about the Switch it seems, safe for Skyrim, but I'll wait and see how many sales this will get. Hard to tell for me. Oh right, Minecraft should do okay no matter where it releases on.

Ubisoft though did already commit to the Switch: Just Dance, Rayman Legends, Mario and Rabbits and Steep. The monopoly game announced for Switch apparently comes from Ubisoft, too.

Warner Bros is in with Lego and Movie adaptations, not too much but at least something already

But I hold any judgement until E3 has come and gone again, as there will probably be a slew of new announcments for the Switch then.

So, after all, the way Nintendo colaborates with eastern third parties seems to be the right one as they all see quite some success. Notice that all of these games above have a significant arcady feel to them which seems to resonate very well with the target audience. All of them don't have the shiny AAA stigma yet still sell good. Therefore, I conclude that Nintendo doesn't need western third parties. They are just the icing on the cake.

I believe many western third parties will use Skyrim as an indicator towards gamers' interest in western games. So this is not in Nintendo's hands, Bethesda is in charge on this front now. Let's pray for the best...