I would prefer still having a kickoff to start overtime as well as after a team has scored.
I think think they should just expand the current rules to allow the team that kicks off to get a possession, even if the team that receives the opening kick scores a TD. After that, the next score wins.
Team 1: Receives kickoff and drives for a TD.
Team 2: Gets the ball, and a chance to answer.
If Team 2 scores, they can still elect to go for 2, to try to win (assuming Team 1 did not convert a 2PC after their score).
If Team 2 converts an extra point to tie it, they kick off to Team 1 and the game is now sudden death.
I think at that point, there has been enough equal opportunity for both teams. If Team 2 gives up 10+ points on Team 1's first two possessions of OT, they have themselves to blame as much as anyone else for their loss, regardless of if they may have been able to tie it on another drive. Sometimes the last team with the ball wins in regulation, that's just how the game goes, and the same can apply to OT without it being "unfair".