Many of the Avengers don't even have powers. Falcon, Iron Man, War Machine, Black Widow, Hawkeye, and Ant-Man just use gadgets or training.
Here's a difference, though. The run-of-the-mill superhero represents a one-off occurrence or isn't even human. Captain America was created from a unique experiment that can't be replicated. The same is true of Scarlet Witch in the MCU. Thor and the Vision aren't human.
Mutants, on the other hand, are born at random. And while most superheroes get their powers in adulthood, mutants gain theirs during puberty, so you have people gaining superhuman powers when they're at their most volatile and don't have the maturity level to handle that. Having seen Deadpool 2, would you want Rusty Collins in a school with your kids? Many parents would not feel safe if that were the case. They'd be screaming bloody murder that someone capable of burning down the entire school on a whim was hanging out in a school with other kids.
There's also the issue that mutants represent the next evolutionary step. Again, run-of-the-mill superheroes are accidents for the most part. Mutants, though, represent a change to human history. They represent normal humans being supplanted and replaced. They represent a possible era where the normal human becomes the minority, and are potentially lorded over by the mutant majority. They're not individual lab accidents, but a whole new race being born, one whose identity is shaped by their nature as a mutant. The irony here is that by hating and fearing mutants, humans are making mutants more likely to place more importance on their identity as mutants. It's tribalism in its purest form, where people on both sides have the deeply held belief not only that there's an "Us" and a "Them", but given enough time, it'll turn into "Us Vs. Them."