Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm.

In any case, I think it's better to read PP as somebody would find a practical, albeit incomprehensible, algorithm for solving NP complete problems.

Although I probably disagree with PP, because even a candidate algorithm that mysteriously works without proof would have practical value, so this case is not predicated on proving.

I think a better example of genuinely practical but rather uninteresting (YMMV) mathematical proofs are proofs of convergence of numerical methods, FEM for example. (I have been through it in school, it was a torture.)