Screw the title and egos, a guy needs to be smart, he is making top money compared to the average joe, in a winning and stable environment, why change that now? Just for power and status? Look around the league there is not a lot of places with a good environment like NE, maybe there's no one. The smart decision is to stay put, the entire league, specially the AFC is waiting for BB and Brady to retire, right now the AFC has 5 playoff spots and 1 bye week spot in the playoff race, year in year out. Stay in the winning boat, when Brady and BB retire than maybe it will be time to make a move as the league will be in a new era.
Also, this is just a thought with no base, but I really think BB did a great job (and still does) in terms of installing a culture in the organization that is to not care about the stat sheet, that's for players, coaches and other people in the staff. If you are doing your job right and helping the team winning on the field, it will be noticed and rewarded.