It wouldn't be odd at all. Just make the X-gene a thing that used to be rare but now post-Avengers 4 has become a widespread recent phenomenon. Older mutants like Logan, Xavier, Magneto have existed before as byproducts of an old era of mutation pre-Thanos where X-gene was more of an anomaly and it didn't occur in humans often so the existence of mutants were something that was easier to be kept confidential(only a few people knew of the existence of them like government operatives and maybe some family members that kept their secrets from public) so that's why we haven't heard of mutants until now because their population size prior to Avengers 4 wasn't big enough to be noticed so they fell under the radar. Xavier, Magneto, and Logan - even though they're mutants they're able to 'blend' in normal society because they're normal-looking mutants but after Thanos is defeated the X-gene somehow becomes more widespread - to the point that some individuals affected by the X-gene start exhibiting more monstrous or freakish physical characteristics(i.e. Beast) and due to a lack of formal education or guidance on how to control and use their powers the more human-looking mutants like Jean and Cyclops have difficulty fitting in regular society thus facing fear and scorn - the recent widespread manifestation of the X-gene prompts Xavier to start a special school to train mutants on how to control their abilities and use them for the greater good.
Much like Spider-Man, Marvel can use the MCU to tell a fresh X-Men story. Marvel's X-Men can explore the contrast between the Avengers and the X-Men and like you said - explore the hypocrisy of mankind. And once you introduce people like Senator Kelly and Reverend Stryker, the social and political commentary will become even more complex. If Stark is still around, I could see Kelly trying to get him to disavow mutants and the X-Men & implement policies that discriminate against mutants.
