I understand that "family" is usually followed by a singular verb in American English, but how about the sentence in the title? Is it ok to say "My family is all doctors." in AE?
The subject-verb agreement in English is not consistent. There is virtually no hard-and-fast rule governing it unless the number of a subject is very clear such as he, she, we, they, etc. When a collective noun is used, it is as much subjective as it can be and it entirely depends on how you perceive the word.
If you think of "family" as individual family members, you should use "are".
If you think "family" as a single unit, you should use "is.
One more thing to consider is the subject complement. If it is plural as in your example (all doctors), it is better to use "are". If it is singular, it is better to use "is".
As the first answer to this ELU question, Are collective nouns always plural, or are certain ones singular? suggests, there is difference between American English and British English.
I like @BarrieEngland's answer to the question, My family is or My family are?:
It would again depend on how the family was viewed. It could be ‘His family were abducted one by one’ or it could be ‘His entire family was abducted while he was away.’
Family is a singular collective noun. So, usually, the singular form of verbs is used with it. In this case, it would be okay to say that "My family is all doctors", because you're talking about your one family.
However, there are cases in which singular collective nouns are used with the plural form of verbs, mainly when members of the collective noun are not in agreement or when they're not necessarily doing the same thing.
Here is a good website with a lot of examples for both cases of singular collective nouns.
In short, both versions are grammatical.
To get rid of the confusion, consider the following (from @keitereth24)
Everyone in my family is a doctor.
All the people in my family are doctors.
and this pair (from @BarrieEngland)
His family were abducted one by one.
His entire family was abducted while he was away.