Even if there are still plenty of issues I haven't read, the mutants of the Marvel universe and their concept has changed a lot over the year. So I'll attempt to make some kind of timeline (other may fill in if something is forgotten):
During the first Lee and Kirby issues, they came up with the concept partly because Lee was tired of inventing individual backstories for each character and how they got their powers. And to make them different he also decided that humanity should fear and hate them instead of worship them as heroes. There was a school for mutants, and mutants were created because their parents worked in the nuclear industry where they were exposed to radioactivity (Sunfire's mother on the other hand was exposed to the raditation from a nuclear bobm). In the early issues we also had a Magneto that considered mutants as superior over humankind and felt they should rule the world.
Both Lee and Kirby were probably heavily influenced by the science fiction at the time. Both about how society treated mutants, and misunderstandings about how evolution worked. There was a time when some believed that it was sudden mutations, triggered by for instance radioactivity, and not natural selections, that was the driving force of evolution. Today we know that both these elements are involved, but the old assumptions must have worked as great inspirations for sci-fi writer back then: https://www.tor.com/2018/06/14/on-th...us-dobzhansky/
With later writers (I have not read the pre-Claremont stories yet, with a few exceptions) the school concept more or less went away. And mutations was just something that happened now and then. Letting every single mutant have one or two parents that were exposed to raditation at work, or during test bombings, would have been limiting. Also the hate and fear of mutants faded away, and they ended up as being pretty much like most other superhero groups, except for the origin of their powers, which must have bonded them together somehow. In those days it was also possible to have a list of all known mutants and their powers.
Days of Future Past is the only storyline in the Byrne and Claremont run where we once again see hate against mutants, and only in a future they succeeded to prevent from happening. The reason why Byrne came up with the story was to prove Claremont wrong when he said that the sentinels were lame villains. So Byrne decided to make them so dangerous that they had killed almost all mutants on the planet, including the X-Men, and turned society into a dystopia. And one of the few ways to do so was by setting the story to a future that never turned real (Claremont would later turn it into an alternative future).
When Kitty first meet Storm and she tells them they are the X-Men, she smiles and ask for an autograph. There are no comments about them being mutants, even before she learn that she is a mutant herself.
Byrne left after issue 143, and judging by the covers on Cover Browser, the type of stories didn't change that much. The first of these issues I read was issue 169 (possibly 168), first published in 1983. This was the first time we are presented to a hidden mutant society, outcasts and misfits, and way too many to come up with individual powers, names, faces and characters. Which would also be the first time for plenty of off-screen mutants, turning mutantkind into a grey mass.
After that storyline ends, things goes back to "normal" for a while. After that the social issues associated with the X-Men starts to emerge. In issue 184 Rachel Summers arrives our time, and she soon learns that all the hate and fear that she knows from her own timeline is about to emerge here as well. She read the minds of other people, and that is the firs time that I am aware that normal humans express hate and fear for mutants. At least I have never read any issues where Xavier read the minds of humans and see the same as her.
By then we already had Secret Wars, which was running form 1984 to 1985, and one could talk about Marvel before and after Secret Wars. Different superhero times had clashed and confronted each other before, but this is the first time we learn that mutants are not fully trusted by other superheroes on a general basis. The healer Zsaji looks into the mind of Johnny Storm, and is able to see who he trust and doesn't. It was probably also in this series that Magneto manifest himself more as a freedom fighter than a villain, and one or two years later takes over Xavier's role at the school for some time. And in 1986 the first issue of X-Factor comes out, where the original five X-Men members pretend to be mutant hunters offering assistance to care of the mutant problem, so they to find and help other mutants.
In issue 185 Storm is hit by a weapon that neutralizer her mutant power. Is this the first time there is a reference to "mutant powers"? That really doesn't make any sense. A mutant was originally just meant to be a superhuman that was born with their powers (or they emerged in puberty) because of a genetic mutation. Their powers are just the same as the superpowers ordinary humans get when they are exposed to cosmic rays, radioactive spiders or gamma bombs. It's just the origin of the powers that makes them different. Are Angel's wings and Nightcrawlers reduced numbers of toes and fingers superpowers? They are the reflection of the mutation, the phenotype. If the mutation express itself in a certain anatomy or powers shouldn't matter much in that context.
In issue 191 it turns out Nimrod has followed Rachel from the future, and is programmed to kill all mutants. We start to see grafitti and hear words like "die mutie" for the firs time. Kitty is sprayed with pepper spray by some of her fully human classmates and almost killed after they suspect her of being a mutant.
In issue 210 we see the first mutant genocide, when the Marauders kills almost all the Morlocks. By then the mutant hating society, which was absent at least in the issues from 70s and early 80s that I have. When you read the X-Men during the Byrne run, which kid wouldn't love to be accapted at The Xavier's School for Gifted Youngsters? Just saw a free sample from the new X-Men, and Scott Summers have certainly turned into a douchebag. I think less kids would want to join the X-Men today.
In the mid 90s we had the Phalanx, which once again proved that mutants were different from other superhumans in more ways than one. They were somehow immune to the techno-organic virus. Then came the mini-series Age of Apocalypse, where "mutant nazis" for the lack of a better word had become rulers of the world. (A character similar to Apocalypse, but which came much later, is Romulus)
In 2000 we saw the first X-Men movie, and the humanity versus mutants became clearer than ever. And it affected the comics. During the Grant Morrison run, we were given Sublime, a sentient virus that is later used to explain why humans hate mutants. Again, becuse they are immune for some reason. Morrison turned the school into the superhero version of Hogwarts. Like the Morlocks, the mutants were simply too many to descibe individually. And we got mutant fashion, mutant subcultures, mutant drugs and so on. It must have been during this time that we got mutants that had no powers, but just had extremely long limbs or looked like anthro animals (and one made of living gas). A very different concept from what we had been before.
At the same time we also got secondary mutations, and a villain like Vargas, that was born with his powers just like mutants, but was not a mutants. Instead he was explained like humanity's white blood cell, and the mutants were the virus.
In addition, Morrison invented Cassandra Nova, which was both a mutant and some mythical being. And The World, where human DNA was spliced with sentinel nanotechnology to create superbeings that could take care of the "mutant problem".
And there was some sort of genetic countdown, which had created literally millions of mutants worldwide, giving us whole areas of cities were mutants lived. And soon all humans would be mutants. Before that happened, there was an even bigger genocide on millions of mutants on the island Genosha. Making the comic more political then ever.
The Scarlet Witch ended up depowering practial all mutants, and it could be in this context that we first learned about "the X-gene". Not "a mutant gene", but a spesific gene associated with all mutants located on chromosome 23, if I'm correct.
What happens after that, I don't know much about. I haven't read most of those issues. The X-Men of the old days is long gone. There are those who says that the comic has always been political and about bigotry, racism and intolerance. No, not always. But it has become such an integrated part of the mutant concept by now, and been given such a huge role and significance, it is no longer possible to go back. If we should see something like it, it would either have to be a "hidden years" concept, or a similar idea about humans born with superpowers for a spesific reason but without being actual mutants.