I think intuitionistic logic is just as valid a foundation for mathematics as classical logic. They both have very attractive features for mathematics. They both appeal to different intuitions about what logic ought to be. When it comes to mathematics, one is more natural than the other in different contexts.

The proof theory of a logic is what is useful to mathematics. It's hard to beat either classical or intuitionistic first order logic when it comes to their proof theories. I think that's why nothing else really catches on. You can build other logics on top of one or the other anyway. Another perspective is that you can encode other logics in them.