The school didn't. They sent her to Yale-New Haven, which is a real hospital, with a real psych ward.
That's where she went through the standard "dangerous person" admission procedure that everyone is reacting to.
If the hospital is an unrelated medical institution, how did her medical history get released to the school? Is this even legal?
Also, IIUC, the psychiatrist who told her "we can't have you here" was employed at the hospital, not the university?
In short, there seems to be some type of relationship between the hospital and the university.
That is what I mean lie and don't trust institutions. Always think what and how they can use what you give them against you. Because sometimes they will.