Oops, fixed.
Plotwise, the origin could be a more sophisticated take on the plot of X-Men #4, where the X-Men fight the Brotherhood for the first time because they're trying to conquer Santo Marco. Replace Santo Marco with the more famous Genosha, at this time just an ordinary island nation northwest of Seychelles.
At the beginning, Xavier's mansion isn't a school, just the mansion inherited from his parents where he lives with Erik. When the news starts to report young teenagers manifesting enhanced abilities, Prof. Xavier and Erik know these kids must be like them. They come up with a plan to meet their parents and invite them to send their mutant children to a prep school for people like them. Erik comes out of retirement to teach history, Xavier calls up his "sister" Raven to teach biology (& power control) and an Army veteran/teacher as gym teacher (his human step-brother Cain Marko? Or would he already hate Xavier? The inevitable Wolverine?).
Prof. Xavier and Erik have to travel the world to recruit their students. Storm is from a (real) African country near Wakanda. Maybe Hank McCoy is actually Scottish. Iceman might be Latino (Roberto Drake).
Erik is an old man who hasn't updated his fashions since he was 18, and wears a purple cape over his suit. The only other "costume" element he dons in the first film is the helmet, when he starts to get up to things he doesn't want Charles to know. The X-Men costume is introduced as blue spandex pants and yellow/blue athletic shirts they wear in PE. Some of them fight in it, but Storm at least is too fashion-conscious to be seen in public in it.
Erik needs a motivation to break with Charles. It can't be that he wants to punish Genosha for enslaving its Mutants, since they'd only have one at the most. Maybe Toad is being abused in a way that's legal in Genosha and Magneto argues that abused mutants should be taken from their parents by force and Xavier argues that they must live in peace with humans, so they can only help after each nation's version of Child Protective Services takes them away?
So Magneto puts on a telepathy-blocking helmet and he and Raven take Xavier's private jet without telling him where they're going. They invite the Scarlet Witch to help them rescue an abused child "like us." She accepts, and they use a weaponized freighter to attack Genosha. The Avengers don't respond, but Xavier deploys the Blackbird for the first time with the X-Men.
There are a few ways you could go with the third act set-piece battle, depending on what effect they want it to have on the status quo. The Brotherhood could just take Toad and try to be on their way when the X-Men confront them in an isolated area in "we don't want to fight" mode like First Class, they could fight all the way to the President's palace but be forced to retreat by the X-Men, or they could actually win. Either way, Xavier and the O5 students return to upstate New York and try to pick up the pieces of their new life.