What does Brady have that no one else has? Hmm... You say not even Rodgers, Mahomes, or Brees has whatever it is that Brady has. I would say that's true. He most certainly does have something that others don't seem to possess.
I think that Brady plays the game with a better understanding of the rules than any other QB. This is in addition to his other obvious plus traits, skills, and talent, just to be clear that I'm not diminishing the man. But he simply does play the game with a set of rules in mind that possibly many players aren't even fully aware of. Look at who Brady has played for. Belichick knows the rulebook as well as he knows his playbook, and why aren't other coaches like that? I mean, both are equally important if you're going to coach a team and call a game. And it's not just about understanding what the rules say but also understanding what the rules don't actually prohibit. Also of significant value is understanding how the referees operate and what they look for and don't look for in executing the rules.
Anyone who watches Tom Brady on the field knows that he is not just a commander of his offense but also a commander of the referees in getting their attention in a meaningful way that will get them let loose a flag and review a play. I've sat through entire seasons of Dolphins and Seahawks football where I didn't see a QB grab his collar in communication with a ref, but Tom Brady surely does it to great effect. In that way, Tom Brady is not just executing the play but he is watching the field for pass interference and holding calls that might go unnoticed and break up the play. Brady is not the kind of QB to let that go when the game is on the line.
Everybody in the NFL should be that sharp. Today's game demands attention to detail.
The other QB who was well versed in the rules was Peyton. Peyton and Tom helped write some of the more important rules that have impacted the game and made it the passing-first league that it is. Just let it sink in.