Tim Gowers wrote:
But I was, and am, more interested in good examples of cases where a proof of a statement that was widely believed to be true and was true gave us much more than just a certificate of truth.
How about Stokes' Theorem ?
The two-dimensional version involving line and surface integrals is "proved" in most physics textbooks using a neat little picture dividing up the surface into little rectangles and shrinking them to zero.
Similarly, the Divergence Theorem related volume and surface integrals is demonstrated with very intuitive ideas about liquid flowing out of tiny cubes.
But to prove these rigorously requires developing the theory of differential forms whose consequences go way beyond the original theorems