If I had to choose one job for him, it would be (a) doctor.

Well, I don't think the first sentence is very natural. A doctor is not a job, it is a professional career. If I could choose one profession for him, it would be medicine.

However, the 'a' can be omitted when you are speaking of the specific job title.

Doctor is definitely a job. (But OK, my question works just as well if you subsititute paperboy or custodian, it doesn't matter.) And "a" can't necessarily be omitted when you speak of a specific job title, for example "he should be a teacher", or "he should become a teacher". So thank you very much, I really appreciate your help... but I ask the same question again.

'Teacher' is not a job title, but 'doctor' is: Dr Suzuki, Dr Seuss. If you think you already know the answers, then there is no point in your wasting our time here.

Hey, I don't know the answer, that's why I'm posting here. But clearly you don't either. The "unnatural" example I posted is directly from "Writing from Within", by Curtis Kelly. It's an extremely famous and well regarded writing textbook, as you probably know as you're a teacher in Japan. Here's the "unnatural" sentence, verbatim:

There are so many things that she would be good at,

I did not say the article was incorrect; you asked why it was missing, and I told you.