They explained it, albeit subtly. When Xavier was speaking (unknowingly) to X-24, he was explaining that the first time he had ever experienced one of his telepathic seizures in Washington was what resulted in the death of the X-Men. Logan endured because of his healing factor. As it was his first one, no one would have had any clue what to do or to even realize that it was Charles.
Logan started dying because his healing factor finally started exhausting itself on the adamantium. Although it's bonded to him (and comics have touched on this a few times), it's technically a foreign agent. Ever since he got it, his healing factor has been trying to "correct" the presence of the metal ever since he received it. He'd actually heal even faster, like Deadpool level, if he didn't have the adamantium. The line where he says "You're the one that put this POISON in me" is in reference to the metal.
As for no more mutants? The reavers at some point mentioned the careful introduction of chemicals and "pesticides" being slowly introduced into the human population like through food, crop-spraying, etc. The whole thinking behind their organization (which I'm wagering is Essex Corp) was to control the mutant population. That charismatic asshole with the sunglasses and the professor that coddled X-24 talked about it toward the end.
Hope that clears it up.
God, what a masterpiece. I thought I'd never cry during a movie, and then I heard the deadliest 8 year old girl in the world utter the word, "Daddy..."
