Am I missing doctor

Unfortunately, the doctor career doesn't seem to work that way. From my experience, you sim is going to be the only one who ever does anything. But, I have seen other medical personnel at the hospital, they just don't work.

When I started my last game my sim had no skills and started very low.. mopping up ect were tasks. I noticed a nurse would go to the bedside and later my sim had the option to treat but not diagnose. So my assumption was the nurse had made the diagnosis. They don't do the same things your sim does they just seem to talk to the patient or even just stand near them.
The big problem is getting them out of the way so you can treat the patient lol. I usually engaged with them somehow to draw them away from the bedside.
Otherwise there is often a doctor running tests most of the day in my game. Some of them spend time on the computer too.

One of the fastest way to get the nurses to move would be to gossip with them about something.