One angle that really hasn't be considered is that perhaps Brady hasn't been in full denial mode because all that he can say is that he never instructed balls to be underinflated, but it is impossible for him to know for sure that the attendants didn't do it on their own. Everyone says there is no way they would take the liberty to do this, but here's the thing: they WOULDN'T do something he didn't want, BUT if what he wanted was a particular "feel" for the ball, without necessarily knowing this was below 12.5, they might have learned to tweak it a bit to get to the right "feel," ESPECIALLY since they knew from experience that the refs sometimes overinflate (see Jets game). The record shows that he at times said he wanted it at "12.5," but he might not have been certain as to exactly whether or not 12.5 fit the "feel" he wanted. They might have known it was hard to have that feel at 12.5, and tweaked it for him--and it is possibly they tweaked it without even knowing whether it was under 12.5, perhaps assuming that the refs had likely overinflated (I remember one report way back saying refs could set it anywhere between 12.5 and 13.5 they wanted--and that is in cases where they made sure to stay within specs, which apparently isn't always the case-- though they generally try to respect QB preferences as requested).
Now, if this is the scenario, he can't say for sure nothing happened, but he can say that HE hasn't cheated, or ordered it to be done. But he can't really give the scenario I describe above without throwing two little guys under the bus, which would look very, very bad. It is true that if this is what happened, they should have all explained it early on, and while it would have been a violation, it would have seemed more minor, especially since it would be coupled with information that the refs themselves were not always on top of keeping balls inflated to specs (overinflated ball). But, there might have been a fear that any degree of admission before the SB would be risky, and once the SB passed, it seemed best to let the league process play out.
I don't think the above scenario can be ruled out.