In the latter, there is no such pattern (there is no such concept as "doctorship").
I don't follow this part.
It seems difficult to reconcile the quantifier (2) with the singular unless we posit ellipsis derived from cases in which "doctors" serves as an object of verbs such as "consult", "call" etc.
The quantifier and plurality/count of the doctors is contextually irrelevant. The focus is on the idea.
One argument against ellipsis is that there are many (perhaps infinitely many) possible sources which all work out equivalently:
Having two doctors is fine.
The status/assignment of two doctors is fine.
The group of two doctors is fine.
Seeing two doctors is fine.
For an analysis of ellipsis in synchronic terms (enough to license agreement!) I would like to be able to identify a specific source. Technically my point here isn't enough to falsify the ellipsis argument (because indeed one particular possible source above could be
the source), but I find it less convincing for that reason.
With "doctors" in the function of subject, this doesn't work:
*"two doctors is fine (for this job)".
I don't follow. "Two doctors is fine" works. And to my ears "...for this job" is also acceptable. Not with something more agentive maybe like "Two doctors operates well", but
actually I think we can stretch that for a metonymic reading and have it work, even though it's a bit odd. It has a sort of comparative sense, like "One doctor operates terribly, two doctors operates well", with a sort of pun reading to it (zeugma?).
In conceptual terms, to me there is a difference between the singular agreement with police/family and the "doctor issue". In the former case, family and police are conceptualized as a collective group containing different elements.
I see "two doctors" as an abstract expression of an
idea, rather than a way of counting how many doctors. It's an answer to "what" rather than "how many".
It's the same situation that comes up in the following:
"What do you want to buy?"
"Two apples is important."
"Three bananas is optional."
"Four oranges is too much."
These are being treated as ideas, like citation forms, presumably being quoted from a list or as hypothetical scenarios. We could relate them to "having two applies" (etc.) but that doesn't seem like a
literal source of ellipsis to me. Instead, I'd say it's more like:
"The idea of two applies is good."
or
"The quantity/amount of two apples is good."
Given that I don't have any way to falsify an argument of the form "it's ellipsis" I don't know what else to say there, but to me I don't think there's anything wrong with having essentially cited elements taking singular agreement, exempted/shielded from external grammatical relations. It's not too different from a direct quotation: "He said 'two apples'."
--
But to be clear, I do appreciate your comments here, and maybe you'll convince me. These constructions are puzzling either way.