Playing stupid has been a well-worn strategy for years because it works so frequently.
As for the cheating part, look at it from their perspective. The NFL has a published rule regarding the inflation standards for footballs. But the enforcement of that rule, by the refs, has been lax for years. Aaron Rogers likes his footballs overinflated - so cut him some slack. What difference does an extra 0.5 or 1 psi make? Brady likes his at 12 or even 11.5. The refs allow it because they know that this has little effect on the game. Game balls show up at the refs room for testing and some or all are either slightly under or over inflated, depending on the QB preference. The refs know this, and they let it slide.
Now someone blows a whistle from off the field and calls the Pats on under-inflated footballs. The refs scramble at halftime to correct the situation that they themselves created by allowing slightly under-inflated footballs into the game in the first place.
In order to conceal the malfeasance on the part of the refs - and the NFL - everyone has to play dumb. Otherwise, the NFL has to admit their part in this tempest in a teapot.
In that case I really think it would be easiest if the nfl admitted that their enforcement was poor. The generally did that when they decided to tighten up defensive holding calls this season.
That seems the most likely answer so far, without invoking invisible Ninja ball attendants.
Perhaps a few thousand people will have learned some basic physics and math out of all this.