There are two things being discussed here:-
1) On-call
2) Shift working
... related for sure; but different.

As many will remember, being added to the On-Call roster was traditionally seen as something of a Right of Passage! However (rather like electrical safety testing), continuing to do something just because we've always done it might not be enough to convince the "managers" of the Modern World.
Personally, I agree that it is generally prudent to have an On-Call system. But (like everything else) it needs to be properly set-up and administered correctly. Large hospitals may need to have more than one tech on-call at any one time, in order to cover the various specialities.
However, many of all will also be aware that "on-call" can also be abused (by both the users, and also - sadly - by the techs themselves). That's why the Manager needs to keep a close eye on it; by, for example, having a debrief following each call-in.
But what about shift work? Two shifts? Three (24-hour cover on site)? 24x7? "Office hours" only? Or what?
As already implied, I am in favour of 24x7 myself. One of the nice things about always having a tech on site is that (once the users have been made aware of that) the tech becomes the first point of call when help is needed. There is only the one number to ring! The tech can go along to take a look at the problem, and either sort it out (advise or reassure the staff, whatever) or, if need be, contact the on-call specialist for the kit concerned*. After all, it's always nice to feel that back-up is available if needed.

*
That was how it was handled at one place I worked at as an Agency Tech. I would go along and take a look, and call for reinforcements if need be (or, as happened on one occasion ... the user wasn't happy with what I advised).