The Avengers are a much harder target than the X-Men in a lot of ways. They're more high profile and have a lot more support from the public and government. X-Men are a much easier target, because up to a point they can have a much easier time getting away with attacking them.
But more to the point, I think it's a case of tunnel vision. Similar to how sometimes certain groups target specific other groups in real life, the purifiers seem to have their focus on mutants specifically.
At the time Marvel did not have the X-men film rights back nor the people in-charge been happy about it. The Inhumans had been on a corporate level meant to replace the X-men but it simply was never going to fly in the mass market due to needing to hold on to copyright. Marvel was always going to put out X-men content. However because they lacked the films and the means to capitalize on them in a way they could with the characters available to them through the MCU there was a corporate push for the Inhumans. There was even for a time a certain person in-charge who made a push so hard that it got him ousted.
This didn't get helped when the X-men fanbase got worried they might be replaced in the eyes of Marvel with the Inhumans. While it was never going to happen, Marvel saw a marketing opportunity and if you'd been familiar with how Marvel and the spider-office marketed Superior Spider-man you saw right through this. Inhumans versus X-men is a marketing gimmick meant to rattle the fanbases into buying the book to support their favorite teams.
So virtually everything around this is built on the corporate need to fill a void, Marvel publishing stoking the fire, a certain person leading the charge and shitting the bed on a franchise, and fanbases getting all freaked out over nothing. It was a disaster everybody thought someone could make money off of.
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The Avengers did have the Superhuman Registration Act. They are select group of people and aren't as powerful or random as inhumans or mutants. The Sentry, Captain Marvel, and a few other can rival the others' power, and even then, they sell them as more controlled. The inhumans and mutants appear out of nowhere and usually don't have immediate assistance to know what's going on.
I think there are lot of different things that ended up giving inhuman vs. X-men (which I haven't nor will read).
Marvel's mandate :
They wanted to push the inhumans. We don't have the official reasons but remember Marvel's CEO veto'd Black Panther and Captain Marvel's movies for years because he didn't believe in Black and female lead. But also remember that many properties have been pushed at one point sidelining other properties. 2099, MC², X-Force, X-Statix, Eternals, Avengers... and so on. It's the way marvel works. They'll try and push brand and see if it works.
Vs. are usual :
We've got FF4 vs X-men, Civil War, Avengers vs X-men... Twice, Shi'ar/Kree, Skrull/Shi'ar, West Coast Avengers vs. East Coast Avengers, Secret Wars...
Marvel's DNA is super heroes clashing with each others, all the way back to Thing and Human Torch. It's Overdone rarely brings anything and should be more mini-series than complete arcs.
Getting noticed :
When you want obscure characters to be relevant, you put them with popular characters.
Clueless :
It's been a while Publishers have no clues how the business works anymore and wander aimlessly trying to fix things.
If your rumor is true, he did push a muslim inhuman girl and an African American inhuman girl as the smartest person in the Marvel universe. I don't know if people realize it, but all of the new inhumans introduced were ethnically diverse or were gay. It might be true on the movie side, and contradictory on the comic side at the same time.
They had two X-Men events (Death of X and IVX) and a company wide X-Men relaunch event (RessureXion). Weren't these events pushing the X-Men at the same time so they could cash in during RessureXion?
The funny thing is that Secret War and Infinity were probably some of their best events recently, and they were hero vs. villain stories.
This is what should have happened. Had Marvel let those two teams work together as opposed to face-off against one another, I truly believe both franchises would have been better off.
It would have been interesting to see the Purifiers or Church of Humanity attempt some action against Asgard, back when it was hovering over Braxton, Oklahoma.
But the bigots have never been terribly consistent with their beliefs. Someone like the Hulk or Spider-Man or the Invisible Woman, not born a mutant, but artificially mutated as adults, should be exponentially more terrifying, since their conditions suggest that *it could happen to you, at any time!* (Unlike being born a mutant, which, you either are or are not, and being a born mutant, in a sense, is more 'natural' than being a Power Brokered super-juice-junky.)
People like Thor, claiming the status of godhood, or using the mystic arts, like Dr. Strange or the Scarlet Witch, or, even worse, suggesting some sort of infernal ties, like Hellstrom, should be vastly more 'hated and feared' than mutants. There are people out there who consider D&D and Harry Potter to be breeding grounds for Satanism. They'd go absolutely nuts over someone like the Scarlet Witch, even if her powers were 100% not magical, and even more nuts over someone like Dr. Voodoo (He's black! He's a pagan! His heathenry not only *works,* but gives him super-powers! Did I mention he's black?).
Meanwhile, despite all this 'hate and fear' nonsense, where the US is totally fine with giant purple robots stomping around suburbia hunting and killing US citizens who happen to be mutants, you can also apparently be a publically known mutant like Warren Worthington, and nobody gives a single **** that you are a billionaire CEO, or tries to get you ousted by the board for being a mutant (or, let's face it, the least-attentive CEO, ever...).
You take less risks publishing a comic book than making a movie and I'm not implying Ike Perlmutter has biases (those are other rumors). I was stressing that those decisions were about money.
They had huge event but at the same time, Cyclops was a villain and has seen one of the most violent character assassination ever, Wolverine was dead and the crossovers were also Death of X (all is in the name ^^)...
Secret Wars had a lot of Heroes vs. Heros and the crappy storyline with the collision of planets were Namor and Black Panther got at each other and so on..
Cyclops wasn't the villain in those events. He was the hero and they made him a martyr. I think they portrayed the inhumans as the villains. They had an event called Death of X, and blamed them. The inhumans were character assasinated. They're hated everywhere to this day because of Death of X and IVX. They made them out to be villains in those stories.