I think Nintendo's software would suffer if they had to go 3rd party. They still make the majority of their money with hardware, so they'd have to become a much smaller company - with less releases that would most likely be limited only to their core IPs (2D Mario platformer, 3D Mario platformer, 3D Zelda, Mario Kart, Smash Bros) due to manpower and cost. You'd never get a Splatoon, or anything that isn't guaranteed to make a profit (kind of like the other AAA developers).
Right now, they design hardware to fit the software they want to make (versus the alternative, making software for hardware they had no say in). It also allows them to make 100% of the profits from their first party software, because Steam/MS/Sony all take a cut otherwise.
The downside is they can't make a console that is going to lose them money on a per sale basis. Because if it bombs, they don't have a conglomerate like Microsoft or Sony backing them up. Hell, even those companies aren't making loss-leading hardware anymore - Scorpio may be a $600 premium product.
It would be difficult for the NX to be worse specs than the XB1/PS4, just don't expect it to be near the Neo/Scorpio. Which is fine because those are going to be premium products and both companies have said they expect developers to continue supporting the original models. As long as NX is comparable specs with x86 architecture and can run Unity/Unreal/etc, there's no reason it shouldn't get 3rd party ports. I doubt they will go with another controller gimmick this time around but there will be something, most likely sharing a library with a new handheld formfactor or some other software gimmick.