I like giving both teams an offensive possession. Both teams get a chance with the ball during the game, but then they sometimes don't in overtime. I don't like that. There have been a few huge games with worn-down defenses where it seemed very obvious that the coin toss winner would win the game.
Imagine baseball said if the 1st team who bats scores, then the game is over. Or in hockey, if your first shooter scores in the shootout, the game is over. Not perfect comparisons, but I still see value in giving both offenses a chance in any sport.