IANAL IANAL IANAL...
It is my understanding that federal arbitration law requires judges to give great deference to the findings and rulings of arbitrators. In many (but certainly not all!) ways a trial judge reviewing an arbitration decision is a lot like an appellate court reviewing a trial court decision. So (again, as I understand it) the judge has to take many of Goodell-as-arbitrator's findings as a given.
However, he does have to determine if the process was fair and if Goodell was sufficiently unbiased. I have to think that to do that he has to look at the facts at some level. I think the idea is that if there is some minimum amount of evidence to back the punishment the judge has to respect the decision of the arbitrator that there was enough evidence of bad behavior to warrant punishment, even if the judge disagrees with that decision. However, if the amount of evidence was so low that a fair arbitrator could not "reasonably" (for what definition of "reasonably" I know not) conclude there was enough evidence of bad behavior to warrant punishment then that can be evidence that the arbitrator was unfair.
So maybe that's part of what is going on?
To piggyback off of this, one thing I've wondered about is whether Berman may use the NFL's lies against them. I can't remember the exact legal term, but there is something about if one party declares something, it becomes part of the requirements, even if it weren't previously.
This was brought up about the Ted Wells investigation. The NFL was not required to use an Independent investigator, but since they claimed Wells was independent, they are now introducing that as their own standard. It was up to them about how to structure their process, but they said they would do it one way, lied about it, and now they want to go back and say that stuff is irrelevant. I do wonder if the judge will look at things like that; in other words, they can do whatever they want in order to determine innocence or guilt, but for fairness, the process cannot just be changed and misrepresented when it suits them best.
Other examples of this: the NFL said they were going to get the scientific results from professors at Columbia, but then they didn't. They said Brady was being punished for being "generally aware" but then his involvement changed for no apparent reason. They said he didn't need to give up his cell phone, then he was punished.
I can see these constantly moving standards as damning to the NFL's case. Even beyond the weak evidence, the NFL has essentially created several standards of judgement, then changed them to "build their case." I think the NFL should have just done an internal league investigation, but they wanted to score PR points by making the public believe this was going to be a fair, judicious process. Now, Goodell is essentially going against the very policies he created. That is the definition of arbitrary. I can see this as very problematic to the NFL's case, and a reason why the judge may overturn the decision.