an example from number theory (where such simplifications are not uncommon), Bertrands postulate

for any integer $n > 3$, there always exists at least one prime number $p$ with $n < p < 2n − 2$

was first simplified by Ramanujan and then later by Erdos who also proved a more general case.

another interesting case study here may be Lindemanns proof of transcendence of PI which is subsumed by later more general results. as Wikipedia states "Weierstrass proved the above more general statement in 1885.
The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these are further generalized by Schanuel's conjecture."

another "possible/controversial" famous/legendary case study here is Fermats Last Theorem; Fermat scribbled in the margin of his book that he had a remarkable proof, but modern consensus is that he must have been mistaken based on the 2020-hindsight of Wiles complex proof. however, strictly speaking, it has not been proven impossible that there exists a short proof.

it seems that later simplifications of proofs is a natural process of the historical/evolutionary progress of mathematics so that results once thought more arcane/inscrutable/complex become more accessible with the polishing/systematization of ideas/techniques.

didn'tthink that the proof could be much improved, but then were proven wrong. $\endgroup$ – David Corwin Jul 9 '12 at 6:027more comments