What difference does it make if they got the ball one time if all they can do is tie then lose by a FG without another chance to tie or win?
I didn't like sudden death that ended with a FG on the first possession but I don't really have a problem with it if the first team scores a TD. The defense just needs to get a stop.
The only way it would be completely fair is just play out the full 15min like it Is another 4th Q. The problem there is ties in the regular season would be more common and as I said, the NFL, the coaches, and players all want OT to be quick.