Hard to say what should be.
I can only tell you what was reported to have happened.
If I call the hospital and ask to speak to my hospitalized neighbor, if he has already left, I don’t think it unusual to say he has already checked out.
Anything more is likely dealing with Hippa.
I think the rules for mental facilities are more strict.
What they could say, if he had checked out, is that there is no one there by that name. Likely, in a situation like a mental hospital is that there is a list of who calls may be received from. Anyone else will be told they can not provide or confirm any patient info. Since it is a mental hospital, the patients probably are not getting a lot of calls.