Are you serious? How about the team with majority possession throughout the game, or the team that spends the majority of its time in its opponent's half of the field? Or most shots on goal? There are many indicators that one team is out-playing another, why not decide on one that seems particularly appropriate? Not only would this avoid the shoot out, but it would probably reduce the number of games that end in a tie in the first place. If a game was tied late, the team losing the tiebreaker would have to go on offense. It could not sit back and play for the tie, as underdogs often do.
You could even do something akin to what college American football now does, a continuation of the game but under rules that make scoring much more likely. E.g., have five offensive player from one team vs. two or three defenders of the other team, give them five minutes to score. If one team scores and the other doesn't, it wins. If they both score, the team that does it fastest wins. If neither scores, the most shots on goal wins.
I'll admit I enjoy watching shoot-outs, but they really do seem to be little better than a coin flip.
Also want to complain about the red card, which I think is cruel and unusual punishment. Why not take the offender out just for a few minutes, as in hockey? Or if he's taken out for the entire game, why not allow substitution, as in basketball? Forcing a team to play underhanded for much or most of the game seems especially unfair in light of how subjective the refs' decisions are. Two players may engage in identical forms of behavior, one gets carded, the other doesn't.