So if someone proves with an algorithm that SAT can be solved in deterministic polynomial time, then P = NP and that's it?
Yes, that would be it – just as if someone had found $a,b,c,n>2$ satisfying $a^n+b^n=c^n$, that would have been it for Fermat's last theorem.
No one ever found such numbers, and the reason turned out to be that there aren't any: the conjecture was true. It seems likely that the reason no one has found a polynomial-time SAT solver after all these decades is also that there aren't any.
There are barriers to proving P=NP. In particular, the famous relativization barrier applies to proofs of P=NP as well as P≠NP. But I don't know whether that implies that a polynomial-time algorithm is hard to find, or just means that any algorithm that you did find would, in fact, not relativize as a proof.
It might also be possible to prove P=NP nonconstructively, i.e., without presenting any particular algorithm. But it's easy to prove that a SAT solver that simulates all Turing machines in parallel until one of them produces a correct answer completes in polynomial time if P=NP, so we would still know a polynomial-time SAT solving algorithm in that case, though it's utterly useless because of the astronomical constant factor.