Ignoring the ref's failure (imo) to allow advantage to develop, the referee allowed Kompany to complain and stay on the pitch for 3 minutes after his red card. Is there anything a ref can or should do at this point, to a guy who is already on a red?
Clattenburg issues the red card (second caution) at about 63:19 and Kompany steps off the field at 64:30. Part of that time is spent him giving his captain's arm band to a teammate.
What to do if a sent off player refuses/takes his time to leave the field is still a good hypothetical one though. If the player is flat out refusing to leave, you can tell the captain and/or coach the game will be abandoned if he doesn't leave. I don't know how well you could sell it, but cautioning the captain if the player is taking his sweet-ass time might work, and always make sure you include the delay leaving in the report.
100% disagree that you can caution the captain. The player leaves or the game is a abandoned. That's all you need to tsll the captain or coach. And the competition authority will come down hard on the player if that happens.
It's really this simple. If the player doesn't leave the field relatively quickly, you inform the team's captain/coach/person in charge that the match will be abandoned if the player doesn't leave in a timely manner. Be sure to note any delay in your match report, even if the problem doesn't escalate to the point where you abandon the match. The competition authority is able to issue additional sanctions to the player for his behavior following the red card, including failure to leave in a timely manner.
Used to have to deal with this all the time in our adult league, as a player who was sent-off had to leave the pitch and be completely out of sight of the field -- out of sight, out of mind. Of course, you would occasionally get that guy who wanted to sit on the bumper of his car and wait for the referee or whoever he felt done wrong by. What I have found works wonders is the power of peer pressure. Calmly announce that if the player doesn't leave (the pitch, in this case) everyone will have to go home without finishing the match. Usually that will get you some crowd support but, if that fails, abandon the match. Not sure the origins of all the cute reasons you might want to caution a captain for the behavior of other miscreants, but I have never ascribed to that theory.
We often say the captain has no special rights; why would we say she should be subject to any special penalties?
It stems from the fact that, while they have no special status of privileges, they do have a degree of responsibility for the behavior of their team. What is that responsibility, one could wonder... But again, I think we are smart to leverage the captain -- or, as I've mentioned before, the influencer on the team -- in a positive manner. Cautioning them for something like this doesn't win you anything.
Maybe he was confused about showing the Captain a card for his teammate who was carried off of the field of play. Captain didn't receive the card, was just shown the card.