Sometimes it’s just dropped as a kind of shorthand, but without any change in meaning. I could say “Road’s closed” in response to someone asking why they can’t go this way, but it’s identical to “The road is closed.” It’s almost like a contraction.
I wonder if some of the uses of “Doctor’s in” are just due to this, as compared to use of “Doctor” as some kind of honorific.
To my children I call the person I married and am sleeping with, “mom”.
Ya’ll worrried about bizarre determiners?
That’s because ‘in hospital’ is generic, whereas ‘in the hospital’ means a specific hospital. It’s no different from saying ‘in prison’ or ‘in school’. I would say the US version is more inconsistent than the British.
Except that while you might be ‘in’ prison, my granddaughter is ‘at’ school if she is physically there.
Inconsistency is what Americans do best 
Not always.
It’s not an honorific. It’s a (generic) name. Fred is ready. Doctor is ready. Mother is ready.
Why not use a full name? Because it’s not necessary.
Fred Nurk is waiting for you.
Doctor Frankenstein is in the surgery.
Mother Hubbard lives in a shoe.
Full name is more formal:
Thomas Sawyer. “That’s the name they lick me by. I’m Tom when I’m good. You call me Tom, will you?
Never really thought of that, but yes… In my usage:
My child is in school = my child regularly is attending school during normal scheduled times
My child is at school = my child is at this very moment attending school
My child is at the school = my child is currently located at the school building for any reason at all (or none) but not for classes
This reminds me of the first line in a children’s book that bothered me when I read it as a kid: “It was Cook’s night out.” https://www.tias.com/tom-and-jerrys-party-little-golden-book-222488.html
Grammar book says (among other things):
“In hospital”, “in class”, etc., are covered under quasi-locative phrases concerning some institutions of human life and society; other examples of zero article are for means of transport and communication, times of day and night, seasons, meals, and illnesses.
I think of “in hospital” as a Britishism. Same with “university”. As an American, I went to a university. I went to college at a university.
You seem to be saying two different things. I’d agree that “the doctor”, in common usage, is just a generic name (aside from “The Doctor” used by Doctor Who). But you talk about a “full name”, which implies that “Doctor” is a shortened version of “Doctor Frankenstein”. But that’s keeping the honorific part, not using a generic label. Or, at the least, using the honorific as a kind of office nickname.
Sometimes the indefinite article gets dropped in casual speech, but I don’t think that’s what’s going on here–it’s not “the doctor” that got elided into “doctor”. Where it’s used, it seems to be used consistently, and possibly as a sign of respect. I wonder if people that use that form capitalize it if in the middle of a sentence.
It’s a first name. Like Fred.
It’s a descriptive first name, like Smith, or Black, or Huxtable or Marshall are (were) descriptive last names. Like Andrew, Mary, Allen are (were) descriptive first names.
‘went to college’ ? Not ‘a college’?
Right, that’s the Americanism I mentioned a few posts back, same thing: “I’m in school/college” means “I regularly attend/am enrolled at a school/college”.
“Went to college” means “I was previously enrolled at a college/university”. We never use the “went to university” construction, and we generally don’t say “went to a college”, UNLESS it was part of a broader informative statement, such as “I went to a college in the middle of the prairie in Nebraska that you have never heard of.”
But that’s exactly the usage we’re talking about;
Went:
to college = was enrolled
to hospital = received treatment
to prison = was convicted of a crime
to church = attended worship
to school = attended class
Both variants of English have these constructions, just not in the same assortment.
I find that to be a very odd way of looking at things. The doctor already has a first name, and it’s probably not Doctor. Plus, plenty of doctors will go by “Doctor Steve” or the like. In non-medical settings, he goes by Steve. In the office, the honorific Doctor is added.
“Doctor Doctor, I’ve got this strawberry growing on my head.”
“Don’t worry, I’ve got some cream for that.”
That wouldn’t ever be used in the US, any more than we’d say, “He went to store.” It’s always “to the hospital.”
Yes, but you do have some of the others such as ‘to church’. Like I said, both dialects do this, just in different assortments.