Why not combine some ideas.
Let's start things off with, whoever gets the ball kicked off to them to start the game, starts things off in OT. That makes deciding whether you want to kick or receive the ball to start the game even more important.
Second, do a college football theme with teams getting the ball at the 40 yard line and each team has one drive to score as much points as they can. The team's both play in the direction of the kickoff to start the game. If a team's defense scores on a turnover, that team wins the game. I chose the 40 yard line so that teams with great defenses could stop the opponents from scoring at all, making that side of the ball important in overtime. It also makes having a great kicker important because he might have to kick a 50+ yard FG.
If tied after both teams get a possession you then go to sudden death with first team scoring wins the game. Quarters are 15 minutes long. The team kicking off, kicks off in the same direction as the kickoff to start the game.
So we make deciding what to do on the initial coin flip important. We give each team an equal one time opportunity with the ball to score in overtime. The game goes to sudden death thereafter.
Everyone should be happy.
