Ironically, (for the question's exceptions), I recall a conversation where it was mentioned the most likely explanation was an in-universe source of what we call "handwavium" - that is, an actual something, some kind of substance, which happens to affects probability. It would cause both good and bad luck, taken to extremes, and increasing the odds for strange rather than logical consequence.
The worlds of the comics would then be the natural results of systems and species evolving in an environment with such a resource, with occasional exposure as a natural hazard (since "both good and bad" is taken quite literally). More chances for "beneficial" changes, by those more directly exposed, less for successful reproduction because they're playing heroes and villains instead (occasional insanity, visible alteration which is not socially accepted, much more dangerous lives, yanno, all that). It probably means strictly selecting for genes which have almost beneficial mutations - since the actual "successes" might not reproduce, but their families have a much better chance. I imagine it would be a little like evolving to live in a high radiation environment, species might have a lot more unstable mutations visible, since anything that can shift around, does - but the resultant species would also be tougher, becoming more likely to survive mutation since any genes which could easily mutate to something irrecoverably bad were selected out of the population. Or something like that.
Or alternatively, other answers mention there should be corresponding numbers of neutral or negative mutations. A very little handwaving might start this off a little higher mutation rate, a little higher fertility rate, and a little higher rate on the non-viability spellchecker that aborts the worst mutations, and the birth rate would look mostly the same. If the overall number of mutations was lower than the other answers suggest, but had been going on the whole time instead of just "when mutants appeared", evolution would eventually kinda select against those genes that would tend to mutate into non-survivable, and for sturdy redundant genetics, and the bias towards neutral and beneficial mutation would build in the population over time. The rise in "useful" mutations might have to do with that timeline, when the kinda-beneficial starts outstripping the neutrally mutating genes (The universe has such beings "occasionally" popping up in history, then becoming common in the era it is set, I think).
There should be a pretty pronounced trend towards variation, visible variation, if so - but given trends we see in our own history, with religious witch hunts, and the rise of scientific skepticism, I could see a concerted and prolonged effort to, um, "select against" visible mutations (aka witch hunt, for minimum few hundred years), followed by a social shift, once it has worked for a while, where society kinda forgot it was really real and not just "superstition" and stories after another few hundred years of nothing too visible, and where occasional oddities hide - which means the appearance of beneficial mutations, is really reappearance of survivable variation once it is no longer being, um, quite so actively selected against. Increasing global migration lets populations of genes mix, and all the recessives pop up, or something - with time enough for useful to get really useful over generations, but a mostly forgotten artificial historical bias against "visible" to explain why most of the people of that universe didn't know/remember/believe all about it until relatively recent in the universe's history.
Which would mean mutants aren't the "next evolved species" so much as a complex and forgotten piece of the whole human race, but then there's nothing quite so human as the tendency to stand on a soapbox and assume they're better, more advanced, and eventually going to triumph and replace those they deem "inferior".