You may not know this, but it doesn't take a lawyer to recognize injustice.
The simple answer here may be just that it wasn't the job of the district court or the second circuit to do justice here. That may be their role in some cases, but it wasn't their job here.
There is a court system. And, generally speaking, its job is to do justice.
But people have decided that there are some disputes they are comfortable doing without subjecting them to the court system. Sometimes people want a system that gives fast, seat-of-the-pants answers, to add commercial certainty to transactions, and they are willing to give up some of the hallmarks of the U.S. legal system that is designed to ensure the most just result. Sometimes, people decide that disputes that humans have with one another should be decided in a way that take into account things other than the ultimate truth or falsity of a factual allegation. For example, parties might decide it's more in line with their economic interests to give a commissioner the right to make decisions about discipline that can take into account whatever he wants to take into account, and even perhaps to punish based on bullshit about "integrity" or on less than 50/50 evidence.
So, humans have invented a
parallel system of "alternative dispute resolution." It recognizes justice as perhaps something to aspire to, but it's not the be all and end all of the process.
So, the law asks, what do we do about this? And your elected representatives passed laws that said, well, we like freedom of contract and so we're going to allow it. And someone said, well, do we want any judicial review at all? And the answer was, well, ok, maybe in the event of really egregious errors, but in the end, we can't have too much judicial review, because then what's the use? We might as well just have judicial review from the beginning. And so the guy who sells widgets and wants some certainty about legal expenses and so agrees to arbitration provisions say, yes, I agree to that, because I can't have every dispute going to court even if that means I have to give up something and might get a much rougher and less robust form of "justice." And the legislatures say, ok, if that's what you want, we'll support that, but don't come crying to the courts when the arbitration doesn't work out the way you thought. And then courts say, ok, so our role is pretty limited here -- let's make sure that the process used was what the parties really agreed to and then let's make sure one of the egregious errors that the legislature has told us we can look at is there. But other than that, we're not going to undermine the law.
And that's your answer. It wasn't the Second Circuit's job to come to a just ruling based on whether or not Brady deflated footballs. In fact, it's ruling was exactly that -- that it was not its job to do justice on the facts. That was Goodell's job. And the players gave him that job.
Condemning the entire justice system because you don't perceive that a court reviewing an arbitration award did "justice" misses the point. You would never use arbitration in a capital case.