As long as intuitive consensus continues to hold, the argument goes, there is hope of a concise logical description of it.

Maybe the concept of "infinity" is a sort of success story. People said all sorts of confused and incompatible things about infinity for millennia. Then finally Cantor found a way to work with it sensibly. His approach proved to be robust enough to survive essentially unchanged even after the abandonment of naive set theory.

But even that isn't an example of philosophers solving a problem with conceptual analysis in the sense of the OP.