I've always thought this was a silly point in TOS, and was glad they fixed it with Dr. Crusher in TNG, who mostly does appropriate medical things, she doesn't hang out with the captain on the bridge. Just another thing, like Kirk constantly beaming down with away teams, that looks silly when you think about it.
I simply liked having McCoy on the bridge. If Crusher had been a stronger character it would've been a crime to keep her down in sickbay.
In universe, at that stage of his career McCoy is likely more in an administrative position that doesn't require him in sickbay constantly.
To be clear, I thought it was great having McCoy on the bridge(and on away teams for that matter) too, he was a funny guy and could give Kirk crap like no one else would.
But even as administrator rather than hands-on, he'd still be running it from sickbay. Plus, I think he thought of himself as an "old country doctor," so you'd think given the choice, he'd be happier dealing with patients than on the bridge dealing with ship-to-ship situations.