It also surprised me how simple the more general proof https://primes.utm.edu/notes/proofs/A3n.html is (except for that pesky invocation of the Riemann hypothesis to prove that there is a prime between successive primes)

But (pedantic): it may be that we use a cannon to kill a fly only because we don't know yet that the thing we attempt to kill is a fly, but it appears we don't know that for sure, either. Maybe it is a bullet-proof fly the size of an elephant (would be a cool result: The Riemann hypothesis is true iff, for all N, there is at least one prime between N^3 and (N+1)^3)

It's because the formula for the maximal prime gap that was used in Lemma 4 at the bottom of page 3, sqrt(x) log(x) / 8pi, is a consequence of the Riemann hypothesis. Otherwise the term becomes x^(3/4 + epsilon) which is a lot larger and messes up the argument. I think it's nontrivially useful here, but there might still be a proof without it