This is by far the most logical, sensible solution to OT I've ever heard, and it's oh so simple that I'm shocked that I've never heard of it before:
http://espn.go.com/blog/nflnation/post/_/id/17241/simple-solution-to-overtime-rules
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How about this, rather than starting 'all over', why not just start the overtime as if the game did not just end," theoldred29 offered. "For example, in the Cards-Packers game, the Cards missed the field goal and Green Bay took possession on the 27-yard-line or so. Why not let them start there? In NFC championship, the Saints had the ball, so they would start where they had it. Same down and distance.
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Brilliance.
You lose a bit of the excitement of the 2 minute drill if the game is tied, as there's less pressure to score quickly in that scenario, but that's the only drawback I can think of. If a team is down and driving, there's still obviously a lot of incentive to score and run your 2 minute drill.