If deflated balls are easier to handle, then it sounds to me like both teams would want the balls inflated to the minimum 12.5 psi. Obviously, inflating them to the minimum that would increase the risk of some balls falling just under that number. Since the refs have two hours to check the balls, one presumes they checked the balls, and those underinflated balls did not enter the field of play. If the refs didn't do their job (and a case can certainly be made for that), then the NFL really does have something to investigate.
I think the worst case for the Pats is their their air pump was not accurately calibrated; I find it hard to believe that the Pats would deflate balls knowing that the refs check them anyhow. It makes no sense.
Further, the delay in the game happened right after halftime and the Pats proceeded to score at will after the problem (if there was one) was corrected.