What are some conjectures, first known to be false through counter-example, but whose subsequent analytic disproofs shed novel insights into the problem?

How about "every continuous function is differentiable somewhere"? Weierstrass gave an explicit counterexample (the "Weierstrass function"). Much later Mazurkiewicz disproved the assertion by showing that the set of nowhere differentiable continuous functions is comeager in the set of all continuous functions. I think it's fair to say that that shed a novel insight into the problem.