I think this is much less surprising today. It's not uncommon to argue that a statement about some structures is true because one can decompose structures into random-like and structured parts ('structure and randomness'), and in either case get to the desired conclusion. Usually one is trying to prove that a certain statement is true for a whole class of structures by this method; there is no one special structure for which we really want to prove it. Take Roth's theorem as an example: we want to prove any positive-density set of integers in $N$ (for some large $N$) contains a three-term AP. We can do this if the set looks random (small Fourier coefficients) easily enough, but if the set does not look random, then there is a large Fourier coefficient, so the set looks structured, namely it has a noticeably larger density on some long AP than on $[N]$. But then we can pass to the AP and iterate this argument.

It just happens that when we try to prove things about the primes, we really only want to know about the one structure, contained in a class of structures with prime-like properties (which might not have more than one member, depending on the properties used in the proof). But the proof method is still to show that the whole class of structures satisfies the desired conclusion.

In this case (G)RH is a statement that the primes behave in some random-like way, in a fairly strong quantitative sense. And so its negation is the assertion of some structure beyond the obvious - which of course can be useful.

Of course, if you want to find other examples of theorems which you can prove by appealing to some major open problem being either true or false, you should probably have some reason why either outcome helps. Quite a few major open problems (or major theorems) can be read as some kind of quasirandomness statement, and for that there is a reason why the conjecture failing might be useful, so in principle there should be a decent list of possible candidates.