Another example is Birkhoff's ergodic theorem.
The modern proof uses Hopf's maximal ergodic lemma, which makes it much shorter than the classical proof is. See, for example, these notes (the proof given here is quite detailed, but not as complex as in the classical approach. I have seen shorter proofs of the exact same statement though, also using Hopf's lemma). It is possible to prove it even more briefly, for instance in this text, where Keane and Petersen prove a strengthened maximal ergodic lemma.
The original theorem stated by Birkhoff in 1931 can be found here, for example. So you can see 'what mathematicians did before E. Hopf proved the maximal ergodic lemma'. I wouldn't call this extremely messy, but it's definitely more complicated.
I cannot give any background as to how Hopf came to proving his lemma, but it must have appeared in his book about ergodic theory, published in 1937. So I conjecture it was inspired directly by Birkhoff's work. (I'd be happy to see comments or corrections concerning this)
