But a 432-state TM is a problem that we would like to "prove" is it not? It's not even a particularly complex one to begin with, my smartwatch has orders of magnitude more state then that and yet here we see that all of our math "fails" at it.

I'm no mathematician, but this is also the crux of Gödel's theorem, he just showed it in a more "hacky" and clever way - but BB(432)'s relation to ZF is also a consequence of Gödel's more general idea, is it not?