No you do not.
You don't want an incredibly tired person prescribing treatment for you or, worse, operating on you, unless you have some kind of death wish.
You want a proper call rota with the doctor "on call" and only the doctor "on call" to have their work phone ON and be available during their on-call hours.
That idea of yours would be perfectly fine if it was just you, but it isn't: it's you and all other people who think like you (or if they start by not thinking like you, they'll change their minds when they see others who do think like that get prompt service whilst they themselves do they not).
That logic just leads to people who if they make a mistake can kill you or give you a problem for life (by prescribing your the wrong medicine or, worse, cutting the wrong thing whilst operating) being very tired and hence way more likely to make mistakes.
Having a single professional having to be on call 24/7 is very much a Tragedy Of The Commons situation - fine if only one or two people used that availability only for very urgent problems, a disgrace for everybody when lots of people innevitably use that availability for any shitty shit little thing.
So the Illuminati created Pluto as a distraction from Autism.
That makes sense.