There have been many suggestions for tie breakers in the past, including a points system similar to the one you suggest. I was a big fan of the "overtime" rule (as I called it) like Lord0fWinter refers to, at one time. However, I am currently on board with the current tie breaker system, and I'll explain why.
You mention that you don't like that the game turns into a damage race during the final 10 minutes. What you seem to be forgetting is that the ENTIRE game is a damage race. Whether you take the slow and steady approach, or the blazing fast approach, you are trying to kill your opponent faster than they kill you. Ergo, a damage race. A time limit, no matter how strict or relaxed that limit may be, simply defines the pace at which the race can be ran. There's no way around this.
So, the question isn't about whether we want to change the game into a damage race, because it already is. Instead, we have this relevant question:What is a fair way to judge the damage race if time runs out and both parties have failed to complete it? I believe it boils down to three options.
1) Award victory to the person in the lead. This is the current method of judging it solely based upon health remaining and is the simplest method with the least room for controversy. If we compare it to a foot race, whomever is furthest ahead after a set time, wins. Clean, simple, and fair.
2) Award victory to the person most likely to win given an infinite amount of time. This method requires a lot of speculation and calculation, not all of which can be conducted in a completely unbiased manner. It could therefore stir a lot of ill will and controversy. How much controversy is a matter of debate, but I think we can all agree it is not as black and white as the first method.
3) Forget about who would have won the race all together and award victory based upon they way they played the game. This is probably the category that most points systems falls into, although such systems could be in the second category. Again, it is subject to a considerable amount of bias, however, as we determine what is worth points and how many points it's worth. This very noticeably affects game play as many players will then structure their builds to maximize points in addition to, or instead of, focusing solely on killing their opponent. I believe this shift in focus would negatively impact the game.
Thus, I am in favor of the current tie breaker because it is simple, easy to judge, and is the most unbiased. It KEEPS things as a damage race, which is how they begin and how they're meant to end. All it does is look at who's in the lead, who's winning, when time is called. It is true that that guy may have been recently tripped and is currently lying unmoving in the mud, but he's still ahead in the race and that is a lot easier to judge than trying to determine if he'll get back up or not.