From what I've gathered on this, which isn't much, both Tom Brady and Peyton Manning after the 2006 season, were vocal and instrumental in creating rules changes regarding handling of game balls, such that the way that footballs are handled now exists. My initial thought, and I wrote it somewhere, was the rule as it exists today - was some archaic rule that dates back to when football teams actually had tight budgets and the numbers of balls provided by each team was something that was actually considered - much like tight budget youth sports. that was incorrect - the rule exists today, because of lobbying led primarily by Tom Brady and Peyton Manning after the 2006 season. NFL QB's do indeed like to have control over the condition of the balls they handle, and for the past 7 years, they've had more control and influence to do that, than ever.
My other position on this, I still hold firmly that I wrote about before. The Patriots, comparably across the NFL, have enjoyed very low rates of fumbling since 2006. There is no doubt that ball that is inflated at 10.5 lbs pressure or less, is easier to squeeze and hold onto, than a ball that is regulation pressure. What seems to have begun by the whims of the QB in handling the ball, seems to have had greater effect, and whether or not the effect on fumbling became a purpose for the intention of deflating game balls or not?
Debateable, it could have been entirely unintended and unknown consequence - of cheating the rules by the QB for his own purposes. But I find it very hard to believe that nobody internally for the Patriots put the two things together
- a deflated ball and decreased fumbling.