NFL overtime rules proposal

I haven't heard this idea floated around, but I think it may be a better way to determine the opening possession than a coin toss. Instead of starting the overtime period with a kickoff in the sense of the beginning of a half, we could have overtime be a direct continuation of the fourth quarter, the same way the second and fourth quarters begin. This way, there is no arbitrary coin toss that often determines who wins in overtime, and teams that couldn't put together a game-tying drive until the end of regulation would be penalized*, since the other team gets the ball to open the overtime period.

UPDATE AS I WRITE THIS–I just realized that the following situation would arise eventually: during a tie game, a team would advance the ball slowly enough that their drive could continue seamlessly to the overtime period. This would take eliminate the drama from some game-winning drives, e.g. the Patriots three Super Bowls were ended with field goals to break a tie. My proposed format ultimately reduces the need for a "two-minute drill" offense.

I welcome comments on this half-baked idea. I would rather have an NFL-style overtime over college football's format, but the opening possession needs to be better resolved. Home field advantage, or some concessions to the team that starts on defense?

*This idea came to me when thinking about the Bears weak but successful comeback against the Saints today.