I’d take it the OP’s suggestion that making it easier to grip ball in the poor weather would have helped the two teams equally—as they were both playing in cold weather. (And actually presumably Indy more, since they threw the ball for a far greater portion of their offense.)
It would not help them equally if the ball boys only put the underinflated ones in play when the Patriots are on offense.
I have no idea whether it happened or not, but Belicheck is a known cheater, and the league is investigating it surely for that reason.