When 3rd parties moved to Sony, it resulted on the PS1 defeating the N64 and Nintendo losing their dominance over the market. This was even more clear with the GC, that suffered greatly against the PS2.
The PS3 was a clearly misstep, so MS managed to grab an equal 3rd party support. That was the difference that made both 360 and X1 do much better than the OG Xbox. Nintendo found success by tapping on a new market of casual players, so they managed to make the Wii a success independently of what 3rd parties thought.
After being a success, the Wii got a healthy 3rd party support until Apple and Google stole that "blue ocean" for themselves. That coincided with the nose-dive on Wii sales.
The Wii U started with a decent 3rd party support. But guess what, gamers who do care about those games don't want to play 30% of them. They want to be sure that they will get ALL 3rd party games. The solutions it to buy a PS4 or X1. The sales of these games on Wii U were damn weak, because the audience moves to Sony and MS some gens ago.
That's why all Nintendo home consoles (except the Wii) sold poorly after the N64. They lost a big part of the market. So they either have to get all (and I mean 100%) those games back on board or they have to find another untapped potential lying unnoticed. The latter is more akin to expecting to win the lottery more than once, so it's not a decent strategy.
Handhelds did not suffer with the lack of support since they are secondary devices to complements a home console. People that have a 3DS are either kids or gamers that already have a PS or XB. Anyway, 3rd party support on handhelds is restricted to spin-offs and inferior versions, so it doesn't matter that much anyway. The same guys that stole the market that bought the Wii also killed the handheld market.
For home consoles, Nintendo would have to regain 3rd party support. That would mean make ports cheap and also pay to force publishers to support its console at least initially. They would have to make a more powerful console that would be basically a PS4/X1 clone. It could still end up with no 3rd party games and be a massive failure since the price would be as high as PS4 and X1.
Switch is a handheld. It's not an hybrid. It's a beefed up 3DS, so expect the same kind of games plus some eventual 3rd party effort or older game port. To compete with smartphones, they added stuff to this handheld that isn't available on phones: packed-in local multiplayer and big screen TV gameplay. They saw 3DS was getting beat by phones, so they added functionality to differentiate Switch from them. As it can be played on TV, it also allows them to keep a foot in home consoles.
So in the end, people will buy PS4s and X1s and maybe get a Switch to play on the go if it is dirty cheap to justify buying a second platform. Japanese market will be an exception since Switch has the potential to be the main platform there. Don't count on a lot of games because of it, since the PS360 era it is clear that the Japanese developers can't be competitive with western devs anymore since they are still stuck on PS2-like games.
Edit: Before people say that there are Nintendo fans that game only on their platforms, I know that. The problem is that the Wii U showed that they are probably a bit more than 10m users, so they won't hold a platform on their own.
