I think a short answer is we don't have a supply of alternative models for prime numbers lying around that make very different predictions for gaps.

If a proof was found of larger prime gaps than predicted, or any other property of primes counter current predictions, I think analytic number theorists would look to the method of proof for clues about a revised random model of the primes. For instance if the proof relied crucially on a newfound correlation between the primes and some arithmetic function $f(n)$, a random model might begin by modeling $f$ and then viewing the primes as random variables depending on $f$.

In particular, for gaps so large that they imply a failure of RH, the first approach tried would surely be to look at the zeroes of zeta off the critical line and how they might be distributed, and use those to get an estimate for the density of primes in a certain region.

proved, not an indication that it is in some way a reflection of the true order of magnitude of the error term. There is no claim that the error term can't be $O(\sqrt{x}\log x)$ or even $O(\sqrt{x})$; it's just that nobody has ever shown something like that. In contrast, the exponent $1/2$ in the $\sqrt{x}$ piece of the error term is known to be optimal, since shrinking that would lead to a known false result (because there are nontrivial zeros with real part $1/2$). $\endgroup$2more comments