It's funny you should mention that, because I was going to say that I think the sport that has it exactly right, when it comes to overtime, is baseball. When you go into extra innings, the game continues on exactly as it has before; no rule changes, no coin toss, no fuss. Both teams get a chance to score, and if at the end of the extra inning, the score is still tied, the game continues as before.
The problem for football is that if an exact analogue of this system were implemented, it would mean playing an entire quarter every time a game went into overtime. If the game were still tied at the end of the extra quarter, yet another quarter would be played. The question would then be raised: how is possession determined in overtime? I see three possibilities: 1) The game continues exactly where it left off, as it does at the beginning of the second and fourth quarters. 2) Possession reverts back to the team who received the opening kickoff in the first quarter (i.e., you open the quarter with a kickoff). 3) You flip a coin for it, essentially starting a new game. I think I would favor the first option, but if an entire quarter were to be played anyway, it wouldn't be all that important.
At the very least, both teams should receive a kickoff. The current rule is so strained: "Well, if a team recovers its own onside kick, the game reverts to sudden death, because both teams had a
chance to touch the ball, right?" I hate sudden death.