Brady has said he prefers his footballs to be underinflated. Aaron Rodgers likes them over-inflated. As the ones who handle the ball the most during any game, quarterbacks have their preferences for a reason. And you can guarantee that centers also have their preferences as well.
In my playing experience, I could easily tell the difference between ball types, sizes, weights, etc. (just as I have preferences for certain weapons on the range, btw). And I knew which balls I could get better reactions with from the grip and feel and would adjust accordingly. Similar to basketballs as well. 1-lb of PSI will definitely alter how a basketball bounces when dribbling or how a shot comes off the rim. In football, the same. In my case, tighter grip meant tighter spiral and a harder throw into traffic with less chance of bouncing off the receiver.
Even at my amateur level, I was much more confident and accurate with a softer ball. Fortunately for me, our practice balls were so horribly ragged and bloated (think medicine ball) that picking up a game ball was like shedding ankle weights or popping the donut off the Louisville Slugger.
With a football, at the skill level of these quarterbacks?... sorry, but 2lbs is huge.
Yet it seemed to hinder rather than help...
Under inflated:
Tom Brady in the first half: 11-for-21, 52 percent completions, 95 yards, one touchdown, one interception, 60.6 passer rating
After the refs re- inflated:
Tom Brady in the second half: 12-for-14, 86 percent completions, 155 yards, two touchdowns, zero interceptions, 157.3 passer rating
Patriots in the first half: Outscored the Colts 17-7
Patriots in the second half: Outscored the Colts 28-0