I would be happy with the following:
Have a kickoff at the beginning of each half, this way each team gets one kickoff.
As for the rest of the game, after a team scores, the other team gets the ball at their 20 yard line and play begins from there.
This way, the kickoff can remain a part of the game and the issue of kickoff-related injuries can be addressed by having only two kickoffs per game instead of after each score.
Oh, and the 4th and 15 idea is just plain asinine.