Is BPP vs. P a real problem after we know BPP lies in P/poly?
Not sure how much of an answer this is, I'm just indulging in some rumination.

Question 1 could be equally asked about P $\neq$ NP and with a similar answer -- the techniques/ideas used to prove the result would be the big breakthrough more so than the conclusion itself. 

For Question 2 I want to share some background and a thought. Pretty much all the techniques and ideas we have for BPP=P, as far as I'm aware, go via "derandomization": Given any probabilistic polytime Turing Machine, construct a PRG to feed it a bunch of deterministically chosen bits instead of random ones, such that its behavior is very similar to its behavior on truly random bits. So with good enough pseudorandom generators, we get BPP=P. (Goldreich's "World of BPP=P" gives evidence that any proof of BPP=P must equate to this.)

This is pretty much along the lines of BPP $\subseteq$ P/poly, except there, the PRG is the advice string which is produced by magic. Perhaps the best answer to your Question 2 is that in P we have no magic and must come up with the advice string ourselves. Derandomization is also the idea behind the 2004 result SL=L, using tools like expander graphs.

Now consider what such a proof would imply for just one particular algorithm, the Miller-Rabin primality test. It would show the existence of some deterministic generator that picks out a sequence of integers to feed to the Miller-Rabin primality test such that, if and only if all the integers passed, then the original number was prime.

As I understand it (though I am no expert), the question of whether such a list exists and how small the numbers in it can be (in particular if it sufficices to check all the numbers below some bound) seems quite a deep question in number theory and is closely related to proving forms of the generalized Riemann Hypothesis. See this question. I don't think there's a formal implication here, but it doesn't seem like something we expect to get next week as an accidental miniature corollary of a much more general PRG construction.