The first two pumps to inflate a ball do nothing to change the air pressure. It's the same inside the ball as outside the ball. Just think about it. The PSI gauge would say 0 (zero), because the air pressure in Foxborough might be 15 psi outside, and therefore 15 psi inside a football with very little air in it.
Eventually the air inside the ball becomes constraint by the leather shell, and the leather starts to stretch. The pressure inside the ball builds. Where it started at 15 psi, it now passes 16 psi, 20 psi, 25 psi, etc. In each of these cases, the gauge would show 1 psi, 5 psi, 10 psi, respectively (just subtract).
The last two pumps to inflate a ball are the toughest. That's where you can get the needle on the pressure gauge to jump the most. It's also at this point there is the smallest increase in actual air required to move the needle. The ball is already stretched thin! It takes a very little bit of air to increase the PSI inside the ball significantly by this point. The weight difference of that extra air will not be seen on a scale. In fact, the weight of an empty football is identical for all intents and purposes to an over-inflated football. It's just air!
While the pressure in Foxborough remains at 15 PSI, the last pump would increase the air pressure inside a ball from 26 PSI to 28 PSI. That means the gauge would increase from 11 PSI to 13 PSI. The air pressure inside the ball increased by 8%, not 18%. If you raise a flag on a building from 5 feet to 10 feet, you didn't double the height of the building, after all (100%), you might have increased it only by a tiny fraction of 1%.
Therefore, if the referees want to measure footballs at half-time -- 4 hours after their preparation begins -- and punish a team if it's beyond a range, then +/- 0.5 PSI is too strict.
Teams bring their own gameballs that have been modified over months (scraped, brushed, polished, weathered, water-boarded, and used in practice) while the manufacturer's guidelines reflect the PSI of a brand new ball. Maybe teams should be given leeway to do what they already do to control the inflation. And maybe the current allowance of +/- 0.5 PSI is too strict given how that can change over a 6 or 7 hour period from 2.5 hours prior to the game until the end of overtime.