"examples where someone proved some (important?) theorem, then (much?) later, someone else rediscovered the insights that led to the proof and built a new theory that eclipsed the original theorem."

According to this updated criterion, Grove and Shiohama proved some basic results about critical points of distance functions in

Grove, Karsten; Shiohama, Katsuhiro. A generalized sphere theorem. Ann. of Math. (2) 106 (1977), no. 2, 201–211

and applied it to prove "a generalized sphere theorem" (as the title indicates).

Gromov realized the hidden potential of this new notion, and pushed it through to obtain far more general results in

Gromov, Michael. Curvature, diameter and Betti numbers. Comment. Math. Helv. 56 (1981), no. 2, 179–195.

Gromov's paper is of fundamental importance in modern Riemannian geometry. It is cited by almost 400 papers.