That would, indeed, be a fair way of determining the winner. The reason it's not done is because people aren't actually looking for a fair outcome, and would feel let down by such a manner of conclusion. They're looking for one team to win on the field. Coin toss + on field play = compromise solution designed to combine some level of fairness with a greater chance of on field play being the determinant.
No. It would actually be sensible, if it were all about fairness. Playing a game for longer than its natural time is inherently unfair. It warps the honest result (tie) and impacts the entire league. The league actually understands that, which is why it allows for a tie after just one overtime period in the regular season, and it's shortened the overtime period in the regular season.
The fairest form of overtime, with no uneven odds and no unnatural play, is no overtime at all, with the game called a tie. But the league doesn't like ties.