i used to think (probably still do) the same thing you've typed here, velcro. if it's not an emergency you deal with it yourself, else you go to ER. after a number of years pdoc finally cottoned on and sat me down for "a chat". he said that when i start getting depressed, that i can find myself at rock bottom very quickly. he says that offering contact between sessions is like an opportunity for him to intervene - if he can get me at 50%, then it's a lot better for both of us than trying to help me when i'm at 0. that he'd prefer to get a call from me than to have to prep me through ER and the stupid hospital system.
he says that he doesn't offer contact to every client, and he's only given a very few his mobile number. he said it's his decision who he gives his mobile to - based on whether they need it and also based on how likely they are to respect it and it's also his decision about whether he leaves it on 24/7 or not.
i think Ts take between-session contact seriously and adjust according to their own boundaries and what the clients require. austin-t for example is happy to take txts from me for whatever reason, but says to call only if i'm at the "emergency" end of crisis.