[You're overthinking this; you can't actually possess the ball without first having the opportunity to posess the ball. Team B got their opportunity when Team A coughed the ball up.
But my point is that that is not how "opportunity to possess" was defined in the rulebook. It was defined based only on kicking plays. Nothing in the rule states a fumble is viewed as an opportunity to possess for the other team.
Because the regular rules of possession apply. The only part that's new is the "opportunity to possess," which they spell out here. The new rules cover kickoffs only because all the other rules of football remain the same. If possession changes at any point on any play in OT, that team had possession of the ball. What happens after that doesn't change the fact that both teams possessed the football AT SOME POINT in OT.
I just don't see the last sentence as the rules are drafted. 4(a) requires both teams to have an "opportunity to possess" (with TD exception); 4(g) defines "opportunity to possess" and states that only applies during kicking plays. So, if Team A never had "opportunity to possess" within meaning of 4(g), then I don't see how 4(a) is met.(a) Both teams must have the opportunity to possess the ball once during the extra period, unless the team that receives the opening kickoff (Team B) scores a touchdown on its initial possession, in which case it is the winner....
(g) The opportunity to possess applies only during kicking plays. A kickoff is the opportunity to possess for the receiving team. If the kicking team legally recovers the kick, the receiving team is considered to have had its opportunity. A punt or field goal that crosses the line of scrimmage and is muffed by the receiving team is considered to be an opportunity to possess for the receiving team. Normal touching rules by the kicking team apply.