You don't think players, or even judges, are influenced by the prevailing conventional wisdom that's out there in the community? They're all people, and when one side is clearly seen as in the "right" or the one to "blame" among large portions of the public, it's going to influence decision makers.
No, I don't. The value of PR is blown completely out of proportion in situations like these, where there is no advantage to be had by convincing outsiders that you are on the side of the angels. This is a bitter pill to swallow for fans, who want to believe that their opinions are worth being swayed, but it's true. Neither the NBA or NBPA can leverage fans to their cause, boycotts will just simply just hurt everyone, it's the equivalent of pooping in the same place that you eat.
I guarantee you that each of the owners, players, and agents considers their cause to be righteous, and won't be moved by anything less than a tsunami of discontent that hurts them personally. Judges may be influenced, even if they shouldn't, but clearly the direct arguments and facts of the case will have a much more significant impact, and all
actual stakeholders will have a chance to weigh in for that.
To suggest that the fans and the media don't have any influence here is, in my mind, starkly incorrect. If PR was irrelevant, the sides wouldn't spend so much time on it.
I am going beyond suggesting it, I'm flat out saying that it doesn't matter a fart in a hurricane. There are times when PR matters: elections come to mind as the best example possible, when the public's opinion actually makes a difference. This ain't one.
I think that both sides are spending so much time on it, not because it's relevant, but because they simply believe that it's relevant.