If the clock runs out in regulation with the score tied, turn the clock off, continue playing (i.e., no coin toss and kickoff), and play to sudden death (first to score wins). There, I told you that you wouldn't like it. Now let's kick it around a bit.
Situation A: One minute left. Team A has the ball, is trailing by 3 points, and chooses to play for a tie. If they tie the game (in regulation), their opponent will NOT have to work against the clock -- they can take as long as they like to try to make the go-ahead score. Once the clock runs out, first to score wins.
Situation B: One minute left. Team A has the ball, is trailing by 3 points, and chooses to play for the win. If they take the lead (in regulation), their opponent must tie or go ahead before the clock runs out.
What are the advantages? First of all, by removing the coin toss to start overtime, you remove a random and arguably unfair element of the game. In regulation, each team got to receive once -- fair enough. Winning the coin toss gives such an advantage, why not just use the coin toss alone to decide the winner? Second, you have new elements of strategy to consider (see above). Third, in a tie, the team with the ball at the end of regulation gets to keep it -- why not reward them for having possession at that point?
Feedback is welcome, but be sure to mull it over a bit first.