The first consideration by the NFL with overtime does not involve us, the fans, unfortunately. Games are set to fit into a timeframe for the networks. Though they do get to run more ads, their schedules get messed up with long overtimes. And players are spent after 60 minutes of play, with an increased likelihood of injuries. So although a 15-minute overtime period seems to be the fairest idea, that's probably not going to go over and is why it is not in effect now.
In my opinion there are two problems with the current overtime setup. One of course is that a team could lose the game without ever getting the ball on offense. Imagine baseball going to extra innings and the first team to score wins, without the other team getting a chance to bat! The other problem I have is that strategy and what you are trying to accomplish is drastically altered. There is no point in risking going for a touchdown; all you are trying to do is kick a field goal since a touchdown and field goal now have the same value.
I would say two five-minute periods with teams alternating kickoffs might be the way to go. That, or a minimum of one possession for each team, and if the score is still tied at that point revert to the current overtime rules.
Regardless, I think there should still be some form of overtime. If I'm not mistaken it was put into effect partly because too many teams were playing for the tie, especially on the road, making for some boring 4th quarters that left fans very unsatisfied, and also mucking up the standings.