I also hate the "they were already kicking around" idea. I agree with yours. Some event from one of the previous films scattered phlebotinum all over and it interacts with the x-gene to activate latent powers. This could be the opening of the portal in Avengers 1, or something from Endgame, whatever. I don't know if the Celestials need to be involved but why not?
We could even say that there were a few mutants before, they were just super rare. The phlebotinum is like radiation. There is always some of it floating around. But we just got a huge dose recently. I would buy the idea that Charles Xavier has always been a mutant. He would keep it on the DL, getting into academia, writing papers on obscure topics with fellow scientist Moira MacTaggart. Also, Wolverine's backstory will probably take some form of "been poking around for a long time". Magneto too.
So there you go, the "Big Three"^TM would be among the original mutants.
But now suddenly teenagers are developing powers. Charles and Moira start looking into how this relates to their research. Charles remembers an old friend from way back in the long ago time of the 90s, Magnus. He had mad theories about "mutations"
I personally think that An Alpha Flight movie would be a great way to lead into the X-Men. Professor Xavier and Moira could be called in to consult on mutants. The use of Northstar and Aurora could introduce to us, and plant in Xavier's mind, the idea of using mutants in a para-military strike force. The film could also introduce Wolverine, who should of course fight Hulk.