Well for one thing, who has ever argued that Nintendo DIDN'T need third party support? Certainly not Nintendo themselves, ever.
But for another, Nintendo first party games DO tend to be the software that pushes Nintendo's own hardware. But third party support IS important, and with system sales, you have to believe it'll steadily come back. Wii U had one of the most robust launch lineups in gaming history, the support was there. It just dried up when system sales dried up.
I have! Of course, 3rd party support can only help, but I don't think Nintendo "needs" their support.
Also, currently the support from 3rd Partys is still there for Wii U, well except from EA. The problem is as you said, it's 1st party that sells their consoles, and they went 8 months without a Nintendo made game.