outside of the OT rule of declaring a winner, which is the game is over the instant the game is not tied after both teams have completed a possession. Other than that, it is as the ref stated and they began a new game once OT started.
I think there is not a clear cut and dried strategy with the new rules. I think the HC needs to have a feel for how the game is going, and if the defenses are ruling the day and not allowing anybody in the end zone and it's a game of FGs, I think there's an argument for going first. Trade FGs on the first two possessions and then win with a FG on the 3rd possession.
But your arguments for going on defense first are certainly valid and persuasive. What isn't valid is an NFL coaching apparently having not broached the subject and determined their strategy before hand.