A TD isn't needed to win. The way I remember reading it was that it reverts to the old "sudden death" format after the first two possessions like this:

Team A gets ball and scores TD it's over.

Team A gets ball scores FG, then team B gets a chance. Team B scores a TD on that chance it's over. If Team B scores a FG, then it becomes sudden death i.e next score (FG or TD or safety) by either team wins.

So it won't be that both teams keep kicking FGs.

This is how i remember reading it anyway.

Yes, regular quarters to make teams change sides of the field.

If there isn't a new coin toss after the duplicate field goal tie, going into sudden death, what have they really gained? You are back to a single field goal winning the game. But I guess the other team at least had a chance and an opportunity to win in overtime.

Exactly. I don't agree with the "logic", but in their eyes the main thing was to prevent a first possession FG from winning it outright. Second and subsequent possessions that result in FGs can win it.

I always thought they should do another coin toss after regulation, then play another full quarter of ball. Thoughts?

This rule is silly, and changing the rules for the playoffs is just plain loco.

All because of the Crykings.

This is it

You forgot this scenario:
Team A wins the toss. Team A punts on first possesion.
Team B kicks field goal on their first possesion...
Game over. Team B wins.

True. I also forgot:

Team A wins the toss and gets the ball. Team B safties them on that possession. Game over.

and:

Team A wins toss. Team B onsides kicks and recovers. Team B kicks FG. Game over.

And
Team A wins toss and gets ball. Team B pick 6. Game over.

Boy, that would suck. Lose a playoff game in overtime by a safety.