I think the Switch has the best chance for Nintendo to make good with western 3rd parties. The Wii had quite a lot of support from western 3rd parties. But much of it was in the form of shovel-ware, or party games. Nintendo provided western developers an opportunity with the Wii thanks to it's cheap development costs allowing them to use it as a support structure to help them make HD games. The problem was that the likes of EA and Ubisoft didn't take advantage of this in any meaningful way. They just saw the Wii as a dumping ground for shoddy ports and minigame collections that they could use to make "Real Games for Real Gamers on Real Systems". Nintendo did all they could to make the Wii developer friendly, but western developers were just stuck on their juvenile "herp a derp, we only make hardcore games for real gamers" phase. The Wii U was what really damaged Nintendo's reputation with developers as it was needlessly complicated and hard to develop for, and it's poor sales simply made the effort to support it not worth it for many companies. Nintendo made too many mistakes with the Wii U, and they suffered because of it.
Times are different today. Developers are more mature, and more willing to explore the market. And the Switch is a very easy and cheap system to develop for. Even if you can't port every PS4 game to it, you can at least get a lot of games running on it with no problem. Plus Nintendo's successful marketing campaign and sales means developers would be more willing to take an interest in the Switch than they ever would with the Wii U. So, I think the Switch can get a lot of western developers on board if it keeps up momentum.
Also to those who say "It won't get third parties because it's weak". Remember that the Switch comes support for nearly every modern engine under the sun. If developers can make Nvidia Shield versions of their games, then they can make Switch versions. As long as your console is easy to develop for, then there should be no problems downscaling it.