This is why I like ties in the regular season (and prefer sudden death in OT). You have 60 minutes to strategize how to win a football game. Many times, teams get complacent and just play for a tie. Both teams get 3 TD, kick 3 XP, add a FG, and have regulation end at 24-24. Yawn. Then when a team loses in OT on the first drive, the fan base cries it wasn't fair. Generally speaking, it's not the players that are griping, as they know the rules.
IMO, teams that don't want to settle for a tie (if there was no OT in the regular season) would be compelled to play differently. Go for it on 4th and short. Don't kick a FG and try to get a TD. Go for a 2 point conversion. Even more so if they went back to sudden death OT. Don't rely on a coin toss . . . be more aggressive DURING the game. I think that would make games MORE interesting, not less interesting . . . and it would end up with games being more fair for those people clamoring that OT isn't fair.
The times OT appeared to be "unfair" the past 10 years stemmed more from the lack of urgency and strategy in the first 4 quarters of the game. How about making coaches make strategic in-game decisions and put the players' feet to the fire to have to come through in a high-pressure moment BEFORE a game ever got to OT? (And that's leaving out the part that teams still are allowed to play defense at the end of games.)
As I see it, teams will be less motivated to try to win in regulation in the playoffs. Now they know they are guaranteed a possession in OT, so why risk anything at the end of a do or die post season game? Personally, I don't think that's exciting football. In the old sudden death OT system, teams would consider going for it on 4th and goal from the 3 with 20 seconds to go in the game. Now kicking the FG is a no brainer move, as that team knows they automatically get another possession. If they adapt the "each team gets a possession" rule in the regular season at some point, I would expect more OT games.