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Tyrrell_McAllister comments on Conceptual Analysis and Moral Theory - Less Wrong
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There is an entire chapter in Pearl's Causality book devoted to the rabbit-hole of defining what 'actual cause' means. (Note: the definition given there doesn't work, and there is a substantial literature discussing why and proposing fixes).
The counterargument to your post is that some seemingly fuzzy concepts actually have perfect intuitive consensus (e.g. almost everyone will classify any example as either concept X or not concept X the same way). This seems to be the case with 'actual cause.' 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.