[erik]

If P == NP, then it could be possible that a proof would show that an algorithm must exist without providing such an algorithm. But that approach wouldn't be necessary, actually showing an algorithm would of course also be a proof.

> To rephrase if this is not the case - what value does solving P = NP provide?

P vs NP is a question of enormous practical interest. But it's also a very interesting question of pure mathematics. A proof that P != NP, or a proof of P == NP that didn't provide an algorithm would still be a huge deal in the math and computer science world.