Thinking about the scenario for just a moment, I can see the following dramatic-potential moment:
If there is a crisis in progress, sometimes the CO is required to make a decision that requires the death of one of his/her subordinates. (That scenario was actually the premise behind the B-story of TNG's "Thine Own Self" -- Troi had to directly order the death of holographic LaForge in order to receive her command promotion.)
Compare that to the additional responsibility of a Doctor: I believe McCoy mentioned before that he'd taken the Hippocratic Oath, which requires him to do no harm. That would present a conflict of interest for him to also, potentially, order the harm or death of some of the crew to serve a larger purpose.
Just a thought.