I think TNG had it right with Dr. Crusher. Yeah, I get that Dr. McCoy has a full time staff and doesn't have to treat every injury but that really doesn't answer the question why he is hanging out on the bridge all the time. If he was really meant to be there then why doesn't he have a station instead of hanging over the railing in Kirk's blindside. That would annoy me. I really don't buy the reasoning that he has to hang out on the bridge to make sure the old crew doesn't crack under the stress.
There are thousands of things a CMO could be doing with his time. How about writing papers about the countless new beings you have encountered and documenting the situations. That on top of the administrative responsibility of running a sickbay would be more than enough to keep him busy.