Having things be unprovable in a body of models would make it not a 100% correct body of models. You know it’d be… incomplete. That’s what it means, we’ve mathematically proven you cannot prove everything that is true.

NFLS is about whether a particular claim is testable, and can therefore productively be debated (as in I’m not debating whether there is a teapot orbiting earth). The way you’ve attempted to combine these two ideas is odd.