The whole point is simply that the notion of computability is not related to coming up with proofs or to any mechanical process for arriving at a result based o

The whole point is simply that the notion of computability is not related to coming up with proofs or to any mechanical process for arriving at a result based on a set of axioms. "Computable" and "Provable in ZFC (or whatever)" are just two different things and pertain to different mathematical objects. Functions are computable (or not). Theorems are provable (or not).