Early to mid-century movie
trope, peaking in the 30's and again in the late 60's and into the 70's.
Nowadays they call it "concierge medicine": having a doctor on retainer, available on call for a select number of patients. In mid-century film, Dr. Gruber is the "family doctor", who lurks vaguely on the edges of the
household staff, ranking somewhere over a servant, yet not quite a full member of the family.
Now, there may be some question as to why and how a doctor is needed for a home full of people who are a) wealthy and b) reasonably healthy. Actually, this isn't much of a question at all, since of course families at this level of wealth are invariably decadent and full of secrets. Assuming that there isn't an Identified Invalid or Elder, we're most probably going to deal with:
1. The Problem Woman. She may drink excessively. She may have "nervous ailments". She may be dealing with nameless "women's problems", which range from an STD from her dear departed husband, to a botched abortion, to cancer in a Nameless Place. She might be A Bird in a Gilded Cage, which means that the Doctor might be keeping her addicted or at least gaslit into staying out of trouble.
2. The Troubled Teenaged Girl/Dicey Heir. Younger version of this, she might be taking drugs for fun, as opposed to having them prescribed. Or, for that matter, she's gotten herself a habit, which the family is hoping will go away. Or she's gotten some Woman Problem, or is suffering from a congenital sickness, or doesn't respond to boys. The Dicey Heir is more likely to have been Sent to Sea or to Military School, although he might also be On Drugs and/or have sexual orientation problems (possibly related to being stuck around other males for long periods of time). Expect him to wake up late and major in being surly and not a little cruel.
3. Spoiled Rotten Sick Children They're the inverse of Being Too Good to Live: able to play Muv and Farve like a violin by getting up a few more symptoms, and looking peaked. Or Muv's too much of a literal cuckoo (as in, lay egg, let it alone) to actually raise them, leaving certain sensitive parts of childrearing (like the Talk) to the good, athem, Doctor. Can lead to #2, if done right.
4. The Madwoman in the Attic As #1, on steroids, but you don't see her until the Big Reveal.
Expect Dr. Gruber to be either a garrulous quack, or taciturn and slightly sinister: the fact that he's neither Upstairs nor Downstairs means that he, like the Governess, leads a life detached from both. Expect him to spend a great deal of time alone, doing "research" of some unnamed type and plotting to get named in the will.