For me, there are two lenses I look at this story through.
The first is through Belichick himself. As a Patriots diehard, there is little doubt that Belichick pushes the boundaries of what is acceptable or legal. When he found that officials would no longer allow his DBs to mug receivers at the line, he began to rebuild his team around his offense and, in particular, his decision-making QB. When teams began to adjust to his spread offense, he re-defined the hurry-up offense. When teams complained they needed more time to adjust to that, he began exploiting uncertainty around what is an eligible receiver. And so on.
Finding competitive advantages on the field and in the locker room where others don't (or perhaps can't) is the defining characteristic of his coaching success. As a result, I would argue that no coach has had a more profound influence on the game -- or individual, if you consider what a coach-driven game and league it is.
Secondly, is the league's culture. Competitive drive isn't just about gold medals and hard work. The same thing that made Michael Jordan the greatest player of his generation drove him to do things like bet $10K on golf putts against friends (and god knows what else). It's what made Bird continue to play for four years longer than he should have with a broken back because he couldn't stand to hang it up before Magic Johnson.
The NFL, like so many other aspects of its culture, has a history with gamesmanship here that has rarely been told. George Allen sending spies to his opponents' practices. Jimmy Johnson and other coaches admitting to filming their opponents. And let's be clear: Belichick has been 100% at the forefront of this: manipulating injury reports, leaving the tarp off the field before a team comes to play in the snow. It's nothing a thousand other coaches haven't done before -- but never under the glare of 21st century media given his success.
For me, this is the clearest reason Goodell destroyed the tapes during Spygate and came down so hard on Belichick. The alternative was basically admitting this long, sordid history. By making an example of the league's most prominent coach, he not only sent a message to other teams and also suggested that the kind of behavior was the exception not run of the mill, and the result of one man's transgression.
None of which is to say Ballghazi is legitimate or not -- it may well be. But the fundamental tension here is not whether Belichick is a cheater or not. It's not even whether sordid gamesmanship has always been a part of the game. It's that these kinds of things are essential to the game and its evolution into the wild success and money printing facility it is today.
Some people have a vested interest in not wanting us to believe that.