To me, it wasn't a minor rule infraction. We hear that football is a game of inches, of tight margins, and it's true. One play can make a difference in a game. One better grip on the ball by the thrower, one fractional ability for a receiver to hold onto a ball that would otherwise be dropped, would change the game. Which, of course, is why they cheated. It's why any player or any team cheats. Baseball is perhaps the most notorious.
Mr. Brady wouldn't let them look at his phone or emails. The team completely stonewalled the inquiry. They might as well have said, "We're covering up." With that, and the texts of the guys playing with their balls, the fact that they were, in fact, deflated, the common sense that the quarterback is darn well able to tell the difference, and the pattern of the team pushing past the envelope, I'm sure, gave the investigators the level of assuredness that they needed.