There are many instances of humans in the comics liking/loving a super-powered being until they realize they're a mutant, and vice versa.
It's not really just about the super powers. It's about what they represent. Spider-Man is the result of a freak accident, but little Sally next door is a reminder that my people's (humans) days are numbered. And there's nothing so frightening as a random child waking up one day and shooting lasers out of their eyes or some shit.
It's not wholly rational, but bigots never are, and you can totally see how the dichotomy works between mutants and other supers. Which is why the whole "mutants won't work in the MCU" take has always been bullshit.
To be more precise, mutants represent what they consider an existential threat to humanity.
With other heroes and villains, the fear or derision is dependent on the individual, since their circumstances are so uncommon as to be benign to humanity in general. There's not that many Harry Osbornes out there with the money to become a supervillain, ditto for the accidents/experiments that preclude some of the others (in fact, there is a narrative that suggests that those supervillains come for human hubris and the naivety of using incarcerated criminals as guinea pigs). Same with the alien "gods" like Thor, their mythology clearly delineates who they stand for and there is no variability there.
But beyond that, mutants can be ANY person born in the world, with all the myriad of motivations and intentions. It makes mutation a "through a mirror darkly" situation, where an average human being's own capacity for unfathomable evil is made all the more terrifying (and all the more undeniable) when any one of them could have dangerous superpowers from a fluke at birth. And that naturally terrifies people, especially when some of them are terrorists that advocate for mutant supremacy (which could be seen as karmic chickens coming home to roost, as it were). It extends beyond a mere race and sexuality allegory by incorporating a true existential dread, one that can be seen in the comic to transcend into even marginalized communities (which is analogous to, for example, LGBTQ people not having solidarity with anti-racist activism).
X-Men can feature useful allegories without discounting or cheapening other real oppression for this reason, but in modern times, that allegory may be better served as a reminder of how important it is to view the world with intersectionality in mind and to draw the correlation directly, rather than indirectly or in a tongue-in-cheek manor. Make the mutant prejudice a cautionary tale about humankind repeatedly making the same mistakes in the same ways.
Furthermore, this sort of prejudice also extends to fear of things like the symbiote aliens and most non-Norse aliens in general and is seen in the Marvel universe all the time.
Maybe unpopular opinion: The X-Men franchise should probably start to pivot away from being allegories to racial and sexual minorities since
A. It's no longer taboo to talk about both those things openly
B. The X-Men live in a facsimile of our world, which means actual racial and sexual minorities exist and experience prejudice
I don't know, it just feels weird watching the X-Men films and there's a scene where blond, blue-eyed Rebecca Romijn talks about her "slave name". It feels like co opting at this point.
The films from Fox were exceptionally more heavy-handed than they had any right to be. There is a track that can be taken that still makes the allegories in X-Men relevant, but you have to be very careful about how you write them.
A lot of mutants don't really get much benefit from their mutation. So I think they justify their anti-mutant bigotry because the other superpowered people they see are either humans or villains whereas mutants might just be people you see at their jobs, selling hot dogs or telemarketing or trying to work in an office. So it's easier to sow the idea that mutants are taking over.
Yeah, this is an oft-overlooked factor: not every mutant in the Marvel universe actually develops a superpower, some are quite benign, particularly among the Morlocks that are seen as repulsive even by other mutants (again, intersectionality should be the word of the millennium for X-Men stories moving forward).
I hadn't thought about that enough, but that's another way that the minority analogy in general, and more specifically the "our mutant children will replace us" rationale for in-universe prejudice only existing against mutants, doesn't really work. In the real world, the fact that gay kids can be born to anyone including rich white parents clearly helped public sentiment shift so rapidly in favor of LGBT rights over the past few decades, rather than the opposite.
See the first part of my post. There is a clear difference between gay people and mutants.
But the thing we all forget is that, for all the mutant fear and hate, canonically there ARE humans who support mutants and accept that they are no more or less deserving of dignity than human beings and the supposed danger they pose is no more or less statistically likely to endanger you than traditional human evils, both malignant and benign.