Great introductory post by the OP. It is interesting that most of the responses focus on the appearance/presser points made. Sadly, the OP's points regarding appearance are too old school for many in today's world. It's been evolving over time but the "dress code" in business as well as a sports management/coaching has gone from business (suit & tie, short hair and clean shaven) to "business casual" (dress coat, tie optional, neatly groomed facial hair) to "casual" (dress shirt and pants, neatly groomed facial hair) to "whatever works" (no code at all). Society today is enamored with the later because it is easiest and saves some time in the morning. Does it matter? It depends on the business you are in and the culture of that business.
Belechick has made it easy for some to say "see you can look like a complete bum and be every bit as great as Vince Lombardi." Hard to argue against that given his results. That said, we haven't seen those results yet, and UCF doesn't count...so I would like to see Scott look sharper and hold back the chew in public (I chewed for many years, even in the business world but never during presentations or when representing the business in public). IMO, the scruffiness is just part of being a "buddy" to the players...he's young still athletic and wants them/us all to know it. As I've said in previous posts, I'd also like to see fewer pressers than he gives...no need for hearing from him so often, especially since he rarely reveals anything noteworthy.
The long and short of it...no one cared how he looked when they thought he'd be bringing home lots of W's, but for some, it's showing that it does matter to them. They see a coach with an unkempt or sloppy appearance leading a team that looks unkempt and sloppy--physically, mentally and on the field. I think that's too harsh on the appearance front. I'm more interested in SF getting the sloppiness out of the mental aspects of the game and, especially, on the performances aspects. Again, it is tough to argue against Belechick.
Regarding the far more important comments of the OP, about being coach to the whole team, not just the offense. I agree wholeheartedly that he should avoid the lame comments about being too busy working with the offense to give a damn about the defense during the game. TO to my knowledge never made any similar type comment and I don't think he'd even imply it...though it may have been true. I'd like to see SF avoid that comment going forward....maybe just apologize to the defensive players (just kidding!).