Even accepting the leaps, the false underlying assumption is that if P = NP there must be a proof. Goedel's Incompleteness Theorem's key result is that within any logical system there are truths that cannot be proven.

It follows that we may never be able to prove that P = NP or P ≠ NP, even though one of the two must be true.

Actually, there are problems that are neither provably true/false, nor actually true/false. In fact I think a standard counting argument shows that 99.9999...% are in that category.

Yes but incompleteness refers to things that are true but can't be proven to be true.

I think you're referring to the idea of prospective axioms which can be proven to be independent of existing axioms (notably the axiom of choice). This is another thing altogether.

In the end, knowing enough about something to reduce it to axioms is only the start and not the end of understanding. P = NP wouldn't suddenly trivialize the problems so much as indicate that a polynomial time solution exists somewhere.