What is a combo point? It is a counter or token. Keep this in mind for later.
What is a finishing move? A finishing move is an attack that will kill or KO the opponent. In order to deliver a finishing move, typically in video games with combat you need to push certain buttons in the correct order or weaken the target to a certain point. Wow requires that you weaken an opponent before a finishing move. This is where the token system steps in. CP are tokens that indicate the level of weakness of the target. You can't turn to a new target and perform a finishing move because they haven't been weakened. Now there are abilities that transfer CP and I feel that this is reasonable mechanic wise and flavor wise. However, this is why combo points stay on their targets.
This is also the ideology that makes openers from stealth hit so hard because you have exploited their weakness of requiring sight.
CP are already like holy power just on the target and not on the player so that is a poor argument imo for why they shouldn't be on the player. If they are on the player they will need to have time limit or finishers will need to be weaker because every 3-5 attacks will be a finisher.
I think it would be a neat idea for CP to be on the player with no time limit, but every time you select a new target (including yourself) you lose a combo point. That way you never have a full combo lined up when you start on a new target and it keep the idea of weakening your target before you finish them off.