Great question!
There is a very common argument that says that the X-books never quite fit in well with the rest of the Marvel line, because in a world where lots of people have superpowers, why are mutants so uniquely threatening? Why is it that they are hated and feared while the Fantastic Four are America’s First Family and the Avengers might as well be the GI Joes?
I’ve never found that to be a problem – in part because there are plenty of other superheroes who also come in for the same treatment (see the Hulk or Spiderman vis-a-vis the Daily Bugle), but also because we have lots of examples in our world of inconsistent attitudes from the public.
My thinking has always gone like this: most Marvel supers have the one-off origin story that was de rigeur in the Golden Age of comics: Cap’s Super Soldier serum is lost, Iron Man is self-built, the Hulk is created in a gamma bomb test, there’s only one Mjolnir waiting around to be picked up by someone worthy. And for the public at large, this is at least somewhat comforting because there’s no reason why they’re special and you’re not, it’s a matter of chance, and who knows, maybe one day you’ll win the superpower lottery, and even if you don’t the people who did aren’t better than you.
But mutants break that comforting illusion: there’s an entire group of people out there who get superpowers because they’re born with it, which means that if you don’t have an active X-gene, you’re no longer “normal,” you’re a flatscan. Which means you’re anti-special, a negative result, you’re a muggle ‘arry. So I could see why there would be a special kind of resentment there. Which absolutely should also be the case with Inhumans and their special latent genes which means they get to come out of the chrysalis a beautiful shiny butterfly while you’re stuck as a caterpillar.
But I don’t think that’s enough to explain “anti-mutant prejudice.” And this is where I think history and politics come into it. In both the comics and the movies, governments in WWII plunged a bunch of money into superpowers and almost all of what they got back was eaten up by the war: Cap goes into the ice, Namor goes back to being a non-aligned power, the Human Torch is deactivated, and so on. And between then and now, governments try to rebuild their capacity and keep powers as a monopoly of the government: hence all of those failed attempts to recreate the Super Soldier serum, Weapon X, etc.
But mutants by their very nature arise outside fo the control of the national security state, and that scares the hell out of them. Your Alpha or Omega class mutants are essentially global superpowers, and can’t be controlled easily or at all. And I think the national security state would start to push that picking and choosing phenomenon: Tony Stark is ok because he’s a human and a military contractor, the Fantastic Four are ok because they come out of NASA, but the X-Men are dangerous rogue elements.
So actually, I think a rebooted X-Men would fit quite nicely into a post-Cap 3 MCU. Thanks to some really clumsy handling of the fallout from the Lagos incident, world governments who were already paranoid about superpowers acting independently start stoking public fear of “enhanced” individuals like Wanda Maximoff. Now all of the sudden, there’s an entire sub-species of “enhanced” individuals hiding in plain sight – are they in league with the rogue Avengers? HYDRA? The aliens? The Daily Bugle demands answers!

To me, the biggest problem is how you fit X-Men’s backstory into the history of the MCU. But I think this is solvable, because we have that nice 60s-70s period of covert superheroing where Ant-Man was active. So back in the 60s and 70s, governments start becoming aware of mutants, but they think there’s just a few individuals. So they keep mutant activity classified, Weapon X starts running quietly in the background, maybe some Sentinels get built during the Cold War “purely for contingency purposes.” And then in the present day, when it turns out that the mutant population has been expanding rapidly but from a really small base, and all of SHIELD’s secrets get leaked…
For most of the X-cast, time isn’t really a factor – either for the original team or the all-new folks or the new mutants or generation x or after, they can really come from any time; Wolverine should have been around for a long time, but his healing factor makes him immortal so that’s not a problem either. The two big problems are Magneto and Professor X. Magneto being a survivor pins him down to a generation that is now rapidly approaching triple digits, and Professor X really should be a peer of Magneto as opposed to significantly younger. So we’re going to need some sort of Bucky-like explanation for why they’ve been kept on ice and why they’ve come out of the ice.