What you're hinting at is the fact that proofs created by human mathematicians are not complete proofs but rather sketch proofs whose purpose is to convince mathematicians (including the person deriving the proof) that a statement (like the Reimann hypothesis) is true. Such human-derived proofs can even be wrong, as they sometimes turn out to be, so just because a proof is given, doesn't mean we have to automatically believe what it proves.
In that sense, proofs can be seen as evidence that a statement is true, and since one interpretation of Bayesian probabilities is that they express degrees of belief about the truth of a formal statement, then yes, proofs have something to do with probabilities.
But, in that context, it's not proofs that probabilities should be attached to. Rather, we can assign some probability to a formal statement, like the Reimann hypothesis, given that a proof exists. The proof is
evidence that the statement is true and we can adjust our belief in the truth of the statement according to this and possibly other lines of evidence. In particular, if there are multiple and different proofs of the same statement that can increase our certainty that the statement is true.
The thing to keep in mind is that computers can derive complete proofs, in the sense that they can mechanically traverse the entire deductive closure of a statement given the axioms of a theory, and determine whether the statement is a theorem (i.e. true) or not but without skipping or fudging any steps, however trivial. This is what automated theorem provers do.
But it's important to keep in mind that LLMs don't do that kind of proof. They give us at best sketch proofs like the ones derived by human mathematicians, with the added complication that LLMs themselves cannot distinguish between a correct proof (i.e. one where every step, however fudgy, follows from the ones before it) and an incorrect one, or an automated theorem prover, are still required to check the correctness of a proof. LLM-based proof systems like AlphaProof work that way, passing an LLM-generated proof to an automated theorem prover as a verifier.
Mechanically-derived, complete proofs like the ones generated by automated theorem provers can also be assigned degrees of probability, but once we are convinced of the correctness of a prover (... because we have a proof!) then we can trust the proofs derived by that prover, and have complete belief in the truth of any statements derived.