I'm surprised that no one has mentioned that, in addition to the possibility that the officials didn't actually measure each of the 24 balls but only felt them, it's also possible that more than one pressure gauge was used at different times.
As a former auto mechanic, I'm acutely aware of the difference between different measuring devices. Others have pointed out that measuring something with air pressure in it usually involves a slight loss of air. When adjusting the pressure in a tire it's amazing how little air escaping it takes to drop the pressure by two or three psi (takes a lot longer to put air into a tire, because of the resistance provided by the air pressure already in the tire).
If the NFL really thought air pressure in footballs was a big issue, and wanted to be completely sure they got it right for every game, there would have to be a fairly elaborate procedure: a) under some standard condition (same temperature every time, say 72), every ball gets checked with the same gauge, which has been recently calibrated and is graduated in tenths of a pound; b) any ball that falls outside of the 12.5-13.5 psi range gets inflated or deflated so that it meets spec; c) any problems found on a routine basis is communicated back to the team and possibly to the league so that other groups of officials can be alert to the possibility of this problem in the future. Then either the footballs or a document gets signed off by the officials. If need be, for whatever reason, the balls can be re-checked during the game under identical conditions, and using the same gauge.
Was this done for the AFC Championship Game? I'd be very surprised it's ever done in a scientifically verifiable manner. The officials have a lot to do before a game (Peter King's story last year is useful in documenting just how much). Maybe they just glance at the gauge to make sure the pressure is pretty close. Maybe they don't even use a gauge. Maybe they have four officials checking the balls so that it doesn't take too long, and they each have a different gauge, any of which might differ from the others by as much as 0.5 psi (not unlikely at all, depending on the quality of the equipment they use).
And the reason they might be this casual about the whole process is that the exact air pressure in the ball doesn't matter very much. Certainly not as much as whether it's rainy or dry, windy or calm, cold or warm in the stadium. Many quarterbacks have said they don't really notice, and others have demonstrated they can't really tell the difference between a 11-psi ball and a 13-psi ball.
Completely bizarre that this gets so much attention, and that former players are claiming that Brady's legacy is in tatters. Bizarre.