What you're missing is that the question-as-stated does not necessarily reflect the real world. The real world does need junction tables. A fully-correct answer to the question-as-stated does not. Change the nouns to anything you desire; the stated problem remains precisely the same, but for this interviewer the answer is different. But not for all interviewers - some do want answers off only what was stated, not off what baggage our language and culture applies to those words.
The question is an artificial, massively-simplified hypothetical problem. What was removed in simplification cannot be inferred with accuracy unless it is stated, and depending on the interviewer, there is always a risk that your answer, however accurate, may be "wrong". It's a guaranteed loss (statistically) for the person being interviewed, and thus most likely for the interviewer as well.
You'd either end up with the same employee or hospital records listed multiple times with different doctor ids (lots of repeated data), or with a field in each employee or hostpital record containing a list of doctor ids (searching by or joining on doctor id requires less simple SQL).
I would think the obvious answer to someone familiar with relational databases would be the OP's expected answer - normalize all three tables, then use junction tables to enable many-many relationships.
What am I missing?