One ball is for the kickers.
That is not correct. From the NFL's official rule book...
Section 2
BALL SUPPLY
Each team will make 12 primary balls available for testing by the Referee two hours and 15 minutes prior to the starting time of the game to meet League requirements. The home team will also make 12 backup balls available for testing in all stadiums. In addition, the visitors, at their discretion, may bring 12 backup balls to be tested by the Referee for games held in outdoor stadiums. For all games, eight new footballs, sealed in a special box and shipped by the manufacturer to the Referee, will be opened in the officials’ locker room two hours and 15 minutes prior to the starting time of the game. These balls are to be specially marked by the Referee and used exclusively for the kicking game.
I would suspect that is why you will see a kicker rubbing, kneading and flexing the football before kick-offs. It's brand new when it's given to him.
The rules say that the balls "remain under the supervision of the referee until just prior to the start of the game" before they are handed off to each team's ball attendants. I guess defining what "just prior" means is the key. How much time before the start of the game did the Belichick have to cheat and was it enough time? "Just prior" has to allow time for the trainers and ball attendants to get their set-up in order so what, 15 minutes prior? As stated by the referee consultant during the game, the teams have enough time before the game to scuff up the game balls (not the kicking balls) prior to use. Stick a pump needle in each one as you scuff them up, done.
But for such a supposed "trivial matter", the rule book does spend a seemingly inordinate amount of time on game ball specs...