Belichick may not have known about the footballs, nor is it his responsibility to check on the status of the footballs. But as stinenavy says, as the head coach, he has to be held accountable for the actions of those under him. That would include players, assistant coaches, equipment managers, etc. Basically, everybody on the field or on the New England sidelines on game day.
Brady is the one who came out looking worse after his Q&A with the media. It's highly unbelievable that he wouldn't notice that his game balls were 2 lbs. PSI lighter at game time than they were when he "approved" them. We hear stories of baseball pitchers who can tell whether a baseball is too big, too small, too heavy, too light, seams are too flat, etc. when the average baseball fan can't tell the difference between a good ball and a bad ball. And of basketball players who can tell that a goal is 2 inches too high or too low. And they're typically proven right. A guy like Brady, who's been in the league for 15 years or so and has thrown over 8000+ passes in his career, claiming that he couldn't tell that the balls were 15% below regulation just doesn't pass the smell test.
If he were smart (and I'm thinking that he's not very), he COULD have said "Well, I thought that maybe they seemed a little bit lighter, but I just attributed that to the weather conditions. I certainly didn't think they were significantly lighter, or were below regulation weight or PSI. If I thought they were, I would have said something to somebody". He might be lying through his teeth in saying that, but at least it has the appearance of maybe being a more believable story.