It's not unreasonable for Brady to look around and see the input that other top QBs have in the league on their teams, and to want the same thing in NE. It's also not unreasonable for him to feel unappreciated or put out if he gets the cold shoulder to that idea. It could be argued that it's a little crazy that the greatest - and one of the smartest - QBs in NFL history doesn't get the same kind of input that other, lesser QBs do.
I understand the tendency to equate reasons Brady chose Tampa Bay with reasons Brady left New England, but I'm not sure that's accurate. We don't 100% know what happened, but most reports suggest that Brady wanted to work out an extension in previous offseasons and the team was reluctant to commit to him long-term given Brady's age and the general cap situation.
Once the situation became clear that Brady was going to leave, I think he gave a lot of thought to what kinds of opportunities interested him, and decided he wanted a totally different kind of experience: new conference, warm-weather city, different kind of head coach, different offensive philosophy and scheme, etc. I don't think he's necessarily saying one is better than the other, but his situation in TB is inarguably different, and if it's an experience that intrigued him, this was probably going to be his only chance to experience it.
I think we've seen he had a lot of input on the play calling and game planning - unless the DYJ videos were misleading - I think he likely wanted more input on personnel and frankly, I don't think that was happening, nor should it have.
I haven't seen anything credible on Brady wanting more personnel input. I can certainly imagine that Brady was dissatisfied with the apparent move(s) from a more modern offense to a more traditional ground-and-pound, punt and win with defense approach. Re: DYJ, it's worth noting that a) that was over 10 years ago and b) that was a season where the Patriots had no official OC (Bill O'Brien was in his first year as QB coach / de facto OC and presumably had less authority than McDaniels had now).