If we're going to be using sports analogies, I suppose we could look at a baseball game with a team at bat bottom of the 9th, with the bases loaded, no outs, but down by one run. I'm not sure what the odds are, but I imagine lots of folks would bet on the team behind with the better "board position". A similar analogy could be made with football, 1st down, 3 yd line, 2 points behind. There is an analogue to board position in baseball and football, though maybe not hockey, soccer, or basketball.
Baseball may not have a time limit, but football does. Yeah, you are 2 points behind, but the clock says 0:00 in the 4th quarter, guess who lost? If it was 0:01, you may have gotten a field goal, but the time is 0:00.
I've also seen a lot of hockey games where one team plays so well in the first period, and they start to slack off on the second and third periods, while the team tries to catch up. When time is up, the team that is catching up is down by one goal, thus losing the game. Yeah, the first team had better board position for 20 minutes, while the other team had better board position for 40 minutes, but either way, the second team couldn't score that many points in a fast enough time to warrant a comeback.
With any sport, or video game that has a time limit, and sometimes a score limit to go on top of that time limit, when time is up, the team with the higher score always wins. Just because a team played so well in the last minute does not make that team the winner. If they played so well, they wouldn't have put themselves in such a rut in the first place.
In any timed sport, or game, there is no such thing as woulda, coulda, shoulda. You didn't score enough points, well in this case, the primary way to score is to draw prizes, then you lose.
There are also total knockouts so again, I think the rule of having 2 extra turns for the person who is to be the loser, because he still has prizes not drawn, to take those turns and try to win. If the future dictates that that person is the victor, then that person should take 2 more turns, no more, no less. This is to help those who are about to win by total knockout.
The whole milling deckout thing shouldn't be a win condition when time runs out. Most likely, it would hamper on people using searches and card draws, so that win condition won't be looked at when time runs out. Prize amount takes precedent over number of cards left.