If it isn't an advantage why do teams never elect to play D first?
And it is arbitrary. As you point out, football contains Offense, Defense, and ST. If you win the toss and kick a field goal on your first possesion, how much defense have you played?
They could just as easily use any of a hundred other methods, but they arbitrarily chose to use sudden death. They could add another 15 minute quarter (like b-ball), they could have a field goal kicking contest (like a hockey shoot-out), they could play a "short field" version like college does, they could play "first to 6" like someone mentioned in this thread (Greenberg's idea?).
And I don't understand the "well, just win it in regulation" talk. That goes for both teams. We have to have some way of settling games that are tied when regulation ends (at least we do in the playoffs), it might as well be fair.