I personally think they should play a 10-minute overtime period. Most games that go to OT are played for that long anyway now, given the posession requirement. If the toss winning team can manage to chew up 10 minutes and kick a FG, so be it. That would be extremely unlikely, and more likely is that there would be multiple possessions and some shootouts. It would be both more entertaining and a purer contest that is less dependent on the coin toss. That would still matter, but not as much.
If they're still tied after 10 minutes in the regular season, it's a tie. In the playoffs, they keep playing 10 minute periods.