Do you not get the concept of superheroes hiding their identities? Superheroes living double lives? Telling the public one thing but being something else entirely? That's Superhero 101.
The movies can still work up to being an actual school with a big student body. Nothing's stopping them from doing that. Eventually.
On top of that, if you are really going to complain about how Xavier's School for Gifted Youngsters should be a school, are you also going to complain that the X-men should just have men as members? See how ridiculous that sounds?
The X-men are not the New Mutants...
Okay, but.. I actually love this idea so much. Having the school, or Mansion, start with the X-Men only, and then having it grow/expand over the films as Xavier opens the doors to public enrollment.
X1
X3:

In this instance, I think a young adult O5 would work best. Maybe the original purpose of the X-Mansion was to be a training ground for the O5 (a proto X-Men team). It was a secret program ran by Charles in accordance with SWORD (Hayward).
Maybe the film opens with their final mission: a battle against Magneto at Cape citadel:
Things go horribly wrong, they fail. Charles loses his legs, and Bobby (the youngest member), is severely injured. Magneto escapes. The program is shut down. Scott is the only one left, he's alone.
5-6 years pass
Transition to whoever kid POV is going to be e.g. Kitty/Jubilee. Their powers spike, and somwhow they end up at this Mansion (Maybe Kitty's father knows Charles, knows he's the only one who can help his daughter)
Kitty is the first "student" at what will become the school. She's the agent of change; the catalyst for the school. She's what makes Scott and Xavier realize that they have so much more to offer to the world.
There could also be an element of conquering trauma. This Mansion is a place where the O5, kids, were radicalized and transformed into soldiers. There's a powerful message there in repurposing/transforming the Mansion from a place of internment to an environment of refuge and understanding, for young mutants. So no mutant kid has to go through what Scott went through. I could see Scott & Kitty forming a strong bond.
The school has a much deeper, more powerful meaning.
Anyway, that's how I'd approach it
Yes!
Been wanting Denzel to do a villainous role again after Training Day
Same here! Now, obviously it would be a very different take on Stryker... Less David Duke, and more Louis Farrakhan. But I think it would be interesting to explore nonetheless; A brother so consumed by self-hate, that he projects himself onto mutants. It'd be like if Jesse Lee Peterson finally found a scapegoat to weaponize his own self-hate against.
I could see Denzel killing this role, and really exploring the layers of this character.