Not my original idea, but I think lead + possession = victory is an elegant way to do it. Still gives an advantage to the team that wins the coin flip (team that wins the coin flip can end the game in one offensive and one defensive possession; losing team needs at least three total possessions), but guarantees that both teams have the opportunity to posses the ball.
I haven't thought this through at all, but what if the game is tied at the end of regulation, they just turn off the clock and keep playing? No flipping a coin; if it's 3rd and 10 from the 45 when the clock expires, that's where overtime picks up. If you combine with the rule above, there's still a sense or urgency to score before OT, since in OT you'll need to regain possession to win. But by making it a continuation of the second half, you side step the issue of needing to flip a coin.