McCoy is not an action character. It would almost be silly to see him in action sequences. Can you imagine McCoy in Uhura's position on Kronos in the upcoming movie?
That said, McCoy does seem to have his "action" moment with Kirk in the first nine minutes. Dragged into it by Kirk, I'm sure, and not with the consequences intended, to be sure. I doubt McCoy signed up for cliff-jumping.
We have no idea what the McCoy-Spock interaction is like in the new movie.
The Kirk, Spock, McCoy scenes in most Trek that I can remember weren't action sequences, but had the three debating what a course of action should be or the merits/morals of a situation. They talked frankly like three officers who had served together for a number of years and were very comfortable with each other.
That social dynamic hasn't evolved in the new universe, yet. Kirk and McCoy are obviously friends. Probably good friends. A friendship is evolving between Kirk and Spock. But there probably isn't even mutual respect between McCoy and Spock, yet. The unfortunate thing about this three-way dynamic is there is no time for it to evolve to the TOS point in a series of short movies.
As far as that meaning Uhura has surplanted McCoy as the third of the "big three" goes, I doubt it.