This is not a hard question to answer as the complications only arise when you try to retcon in the entire 56 year publication history of the X-Men's lore into the MCU -- which would be a fool's errand because it only serves to confuse people and create countless more questions in what should be a very simple, easy to follow introduction.
The answer to "where are the mutants" can easily be found in the early issues from the Stan Lee/Jack Kirby X-Men run. In the original run -- prior to Charles forming the X-Men and before the atomic weapons testing started activating the X-Gene -- mutants were a VERY rare species. Charles grew up as one of the few around the world. He was doing papers and thesis presentations on the prospects of the hypothetical "X-Gene" with Moira McTaggart in Grad school as well as traveling the world after serving his time in Vietnam. The mass emergence of mutants was happening relative to when the 60s books were being published, not decades before the first issue in the Marvel Universe timeline. Follow the comics' explanation and tell that story in real time. This is why "Cerebro" was built in the comics, when mutants were still new and still emerging.
The most simple explanation is the best explanation. Charles Xavier, Wolverine, Mystique, Irene Adler, Sabretooth etc are a couple of the hundred or so mutants that exist in the MCU. "Where was Charles Xavier during the Batte of New York?" Simple, he was in Egypt: on the other side of the planet. (He still had his legs and he was traveling the world) Bumping into a then 14 year old Storm who was still a thief living on the streets of Cairo as an orphan:
Wolverine is in Canada with Alpha Flight, Erik is in Bosnia and Raven/Irene are in Europe (These two do not even know they are "mutants" yet).
The events of Avengers: Endgame cause the X-Gene that lies dormant in thousands of adolescents to be activated in 2018-2019. The snap/Infinity Gauntlet causes accelerated evolution in natural selection. The Eternals would establish that mutants are a doomsday clock that's set go off at a certain point (because Celestials) and the snap has just accelerated the process. What should be happening in 300 years is now happening in the span of 5. This is the how you can keep the evolution angle but tie it into the biggest event in the MCU's history. Why are mutants? Because they are nature's natural response to the events of Avengers, AOU, IW and Endgame. As
@sleekstereo said, they are the result of nature's need to evolve humanity in preparation from outside threats that they can no longer defend + the presence of people with powers contributes to the need for mankind to change. Mutants are quite literally --Mother nature's answer to the Avengers.
Normal people would feel left behind in a world of gods, mutants and mutates. "Why do people hate mutants but not the Avengers?" That's another answer that can be found in the original run. How did mutant hatred begin in the comics? Propaganda and fear mongering. In issue #14, Bolivar Trask was the one who started instilling fear into people about mutants. He was writing papers about how mutants are going to enslave mankind, trying to convince people the need of Sentinels
From this point onward, mutants were feared and hated. You can transfer Trask's role to any other politician and the result would be the same. The Government introduces fear and suspicion into the general public because they want an excuse to go after, imprison and experiment on mutants (all of which they'd already be doing anyway, just without legislation). There are many real-world examples of those with a platform introducing the seeds of hate into the general populace for marginalized minorities. So there's your answer to that question.
And the rest is history
