Well, I think facially people understand the problem with the term "generally aware" in this context, even if they've thought about it or not. Brady was disciplined for conduct detrimental to the league. Awareness, and worse general awareness, is not any kind of conduct. It's a state of mind. How can you be punished for engaging in detrimental conduct, solely for knowing something?
Generally thought crime is not a crime. Knowing something is not, under any system of law or punishment, grounds for punishment, and it especially cannot be if the thing you're punished for specifically is for "conduct". For the lawyers, there is the mens rea (your state of mind) and your actus reus (the actual criminal act). General awareness only goes to mens rea, not actus reus, and even for the non-lawyers if they can't articulate why that's a problem, they have an intuitive sense that it's unfair.
Now, what Wells probably meant, and what the NFL could have said, is that being aware of something is conduct detrimental when that awareness gives you a duty to do something. Being aware that, on your watch, footballs are being deflated below the legal limit, the NFL could have said, gives rise to a duty to put a stop to it, and by not doing so, you're as guilty as if you'd done it yourself. That is, your crime is not one of commission, but one of omission, and in fact failing to act, when you have a duty to act, can be punished under the law. To go back to lawyer speak, it's sufficient actus reus.
But the NFL has never argued this. I'm not sure why. It seems like the much better way to go. And, even with respect to what a putz Wells was here and how wired this process is, if you could open his brain, or put him on a lie detector, I think this is close to what he actually concluded, right or wrong, whether he articulated it well or not.
The NFL's tactic, instead, was to try to "fix" the problem with "generally aware" by having the Commissioner make findings that Brady actually went beyond general awareness. Will this be a winning strategy? Maybe. It gave Brady a procedural fairness argument it didn't need to give him, and set him up for good argument. Berman's question gives us some hope that he's not buying the argument that Wells' basis for punishment is irrelevant if Goodell fixed it -- he seems to think, at least for sake of argument, that the "general awareness" standard employed by Wells is somehow relevant to the case.