In regards to the NFL overtime rules, the coin toss should never be more than a random way of determining certain variables. The whole of idea of “winning” a coin toss should be anathema to any sport where physical skill is the main factor in determining the better side.
Before overtime, after the coin toss, the side that guesses correctly should be given a choice. Either they receive the ball but are banned from attempting a field goal during that initial drive OR they kick it off and give the ball and the ban to the other team. In this way, there is a trade-off from getting the first (and possibly only) crack at scoring. The game would then proceed normally after that.
Also, during this first overtime drive, there would have to be rules in regard to fourth down conversions. On fourth downs, since the field goal option is out, the team can either punt or go for it but the latter can only be attempted one time and, perhaps, not past a certain point on the field, such as the opponent’s 40 yard line.
Another alternative is to award the win to the first team that scores a touchdown, or its equivalent, two field goals. That is, the first to get to six points. In this way, if you want to play it safe and go for a chip shot three points, that’s fine but you’re just giving the other team a chance to score a touchdown and win on the next drive.
Of course, this would increase the number of games ending in ties.