But they could do alternating possessions like college.
Sure, but that's sort of like doing penalty kicks in soccer. Not really the core of the game.
While 'fairness' is great in concept, it would only take a 103-102 game that last's 6 hours for that to look bad.
Having stayed up until 2am to watch a 5 ot hockey game, or even the Red Sox/Dodgers 18 inning WS game, longer is not the best way to go.
My idea would be this. 1st OT is a 15 min qtr. Win the qtr you win the game. If tied the game continues into sudden death in a 2nd ot qtr. It would make ot longer but changes the strategy for the 1st OT quarter and get rid of the nonsense of the more complex scoring rules (i.e. TD wins, FG doesn't etc).
And if you really want it to be as fair as possible. No coin flip in OT. Treat it as the start of another 'half'.