If I had to guess - and it's only a guess - I would hypothesize that the root cause of Brady's problems is that the NFL and the owners (a) have monopoly power and (b) are abusing that power. To me, the NFL (as a corporation, and the constituent teams, as a cartel) obviously have monopoly power in the market for professional football.
To make a claim that they are violating the Sherman Act (governing antitrust claims) you have to claim, very loosely, that they are "abusing" that power in some way.
Now, I am not an antitrust law expert. However, my sense from having studied hundreds of antitrust cases (just in outline) is that:
(a) the courts, for whatever reason, allow professional sports to get away with Sherman Act violations that no other industry can get away with, because
(b) the courts believe that there are "equities" that favor the creation of these huge all-powerful leagues, that is, they believe it just makes more sense.
Now, monopolies are not all bad. A monopoly has a lot of power and, like a superhero, can use that power for good. But the big danger of monopoly is that when a monopoly has become corrupted, the monopoly is so powerful that ordinary people - or suppliers, or competitors, or employees - have no recourse.
What I believe may have happened with the NFL is that they got so powerful, and so rich, that they literally became untouchable. They control vast TV networks (one prominent sports analyst who questioned the NFL's deflategate narrative found himself summarily out of a job, for instance). After all, the TV networks need the NFL revenue, thanks to the monopoly. Their legal war chest is unbounded. When someone like Brady tries to go against them, they are just too powerful.
I personally have never seen a private corporation utilize its monopoly power in as clearly abusive a way as the NFL has done to Brady. I've seen and read of hundreds of corporations harshly penalized for far less serious or clearcut transgressions. The NFL's actions, in which it leaked false, damaging information that was broadcast on major networks that were its own contract partners to target an innocent individual; and in which it then conspired to fabricate evidence against that individual; were extraordinary.
And the NFL could not have done those things if it did not have monopoly power.
If I were thinking of pursuing legal action against the NFL, I would look into the antitrust area. In fact, I think that the NFL's behavior during deflategate pretty clearly suggests that the "antitrust exemption" that allows them to control professional football should be removed. They were given unprecedented power by the courts; they abused that power; so the power should be taken away.
Disclaimer: The above are just some thoughts, completely unresearched, based on dimly remembered facts that may or may not be accurate. I am not expert in the relevant areas of law.