I agree about the idea of a therapist allowing something. I do not think of it like any professional allows me to do anything. I would not continue to see a therapist who spoke in such terms. On the other hand I do understand about professional boundaries (like a dentist or attorney or whatever) and know that if any client becomes too annoying to anyone they hire, the person hired may decide to quit representing them.

But I also know I am not available to clients or students 24 hours a day and I do make determinations about when to call back.
If a client is calling me at home or at all hours of the night, I talk to them about it and if it continues, I tell them to find another attorney (the small client base I still have are criminal defendants and people who have been diagnosed with some mental disease or defect - our statutory terms - and they are often quite anxious/confused/angry etc and call repeated numbers of times during a day and through the night. I can easily get 15 calls in a row overnight from a person who has just been placed on an involuntary mental health hold).
With students I do not always answer emails right away. There is a tendency on the part of some students to hit a snag and email right off the bat - so I have found if I wait a few hours or until the next day - they have resolved the problem on their own and moved on. (I do tell them right up front my email policies).

I would not contact either to assist if there was an actual and immediate emergency. First, one of them already has a message about if there is an emergency, to go to the ER. Second, they probably would not get back in time for any emergency to not have already been taken care of by me. I might contact if I thought reporting about the emergency would help me in any way. If there is a super big deal going on, I call and see if there is an appointment I can make. One of the ones I see works during the first part of the week and the other during the second, so I would choose the one that fit in with the part of the week I was in when the situation arose.