Although the other answers point out correctly that the exact value of $\text{BB}(n)$ is independent of ZF for large enough and even moderately sized values of $n$, nevertheless I should like to point out that the main outline of your proof method is correct. This is a legitimate method of proof.
That is, what I am saying is that if we could somehow know the true value of $\text{BB}(n)$ for certain values of $n$, then indeed we could use that knowledge to determine the truth of the Riemann hypothesis and so forth, for any $\Pi^0_1$ statements whose truth is known to be equivalent to the non-halting of a Turing machine with at most $n$ states. And indeed for every $\Pi^0_1$ statement there is such a Turing machine, since one simply writes the code that goes and looks for a counterexample, halting when found. This machine runs forever if and only if the statement is true. For many classic conjectures, the number of states $n$ would not be very large, although the value of $\text{BB}(n)$ would be enormous, as you mention.
What I am saying is that the theorems would be proved in the theory $\text{ZFC}+\text{BB}(n)=N$, using the additional assumption about the particular value of the busy-beaver function. You would be showing that in every model of ZFC in which the busy-beaver number at $n$ has that value, then indeed the Riemann hypothesis would be true (or false, depending on the outcome of the calculation you propose to undertake).
But furthermore, this very argument can be taken as a proof that the values of $\text{BB}(n)$ cannot all be settled in ZFC (if consistent) even for some small values of $n$. Indeed, the value of $\text{BB}(n)$ for moderate $n$ cannot even be provably bounded by specific numbers, for if we could do this, with $n$ large enough to undertake your proposal with the Rosser sentence, say, and then ZF would have to settle the Rosser sentence, which it does not.
In essence, your proposed argument method becomes a proof that the busy-beaver function values must be independent of ZFC or indeed any foundational theory we might want.
A philosophical can of worms. The method also opens up a certain contentious philosophical debate, however, concerning whether indeed there is a "true" value of $\text{BB}(n)$. If the assumed value of $\text{BB}(n)$ is indeed the true value, then you will get the true value for the Riemann hypothesis and so forth. Many mathematicians believe very strongly that arithmetic assertions of this kind do have a definite absolute truth value, whether or not they can be proved in ZF or ZFC+large cardinals or what have you. The independence phenomenon is seen as a consequence of the necessary weakness of our theories, rather than any difficulty for a robust notion of truth.
But meanwhile, according to the philosophical position known as arithmetic pluralism, there may be no definite fact of the matter for some arithmetic assertions. We already know how different models of ZFC and even ZFC plus the strongest large cardinal axioms we have can have different nonisomorphic arithmetic structures $\mathbb{N}$, with different arithmetic truths, and there seems to be no fully satisfactory argument that arithmetic truth is determinate. Perhaps the most convincing argument is the Dedekind categoricity argument, which shows that there is up to isomorphism a unique arithmetic structure satisfying Dedekind arithmetic, which includes the second-order induction axiom. But since this theory makes reference to second-order structure, essentially arbitrary sets of numbers, it would be basing the definiteness of our concept of the finite upon our concept of "arbitrary set of numbers". But how successful can that be? It would be essentially circular.
