I'd rather not repeat the obvious reason why it sucks (or trying to weigh in on the "your defense has to make a play argument", but I've thought about this and I believe my idea actually makes the process more fair..
Here's a few options..
1) Run the full 15 minutes of OT. Whoever has the highest score wins. In the regular season, a tie score would be a tie. If there is a tie after the OT time in the postseason... I have an awesome idea... both sides have to run the 2 point conversion play and take turns. Whoever messes this up loses (when both teams have taken their turn). This is based off the hockey "shootout" idea.
The only real negative is how one could mess with the clock and injury related issues.
2) Both teams are able to run 1 (but preferably 2) possesions. In the regular season, a tie score would be a tie. If there is a tie after the possesion(s), then we go to running the 2 point conversion. The fundamental idea is kinda based loosely on college football OT rules, except college football OT rules are actually stupid (except for the 2 point conversion try starting at the 3rd OT). Rather than allow the clock to dictate stuff (except the play clock, which might need to be adjusted), the teams are allowed to play in fairer terms. There will still be a focus for special teams (you can still do the crazy onside kick if you want) and health should be less of an issue.. and the clock can't be manipulated for another team's favor. This process will still allow 4th quarter FG wins or extended clock killing drivers in the 2nd half... so this should help both parties out...