Nintendo itself would survive just fine as a third party publisher.
However, since they wouldn't have to provide their consoles with games anymore, don't expect them to produce much else than Mario or Zelda titles anymore (plus Pokemon, Yokai Watch and maybe Splatoon and Fire Emblem) as then they will have to make sure their games make a steady profit. Which would already be risky (for their investors, probably not so much for Nintendo themselves) with Kirby, Donkey Kong or Paper Mario and would only be released once in a long while. Metroid, Star Fox, F-Zero, Chibi Robo, Pikmin or any new IP to name just a few? Wouldn't be worth the trouble anymore, so don't expect any of these games to come out anymore if Nintendo goes third party.
Also, the quality might suffer as they wouldn't need to be pitch perfect anymore to make their consoles really sell.







