So to me it is about what function the moral judgment is being used for? How is it playing out in power struggles and claims of entitlement.
Jesus' quote is a good use of this idea as an intervention. A woman is about to be stoned to death for (possibly) having done something that other people around her are also involved in, via direct action or strong desire. He interfered with a very harsh moral code where those making the judgments were going to abuse power. It was not simply their upsetness about her not respecting marriage. It was not a cheated on wife yelling in anger at the woman who slept with her husband. It was the unmaking of a human based on a hypocritical moral judgment.
But to be noted - There is a difference between making a moral judgment, and acting on that judgment in one way or another. Yet I often see that people think they are one, so "He that is without sin shall cast the first stone" is applied also to deter from making a moral judgment at all.
I make the moral judgment that eating meat is morally wrong; but I do not persecute people for it, I don't personally criticize meat-eaters. But given the chance, I would vote against building a slaughterhouse, or against a political candidate who supports meat-eating.
One problem I see with the model that one must be good to judge others is that those who are more self-aware are less likely to feel justified.
Absolutely.
If you can sense your own potential hypocrisy around an issue, can make connections between apparantly different but at root similar moral acts and deeply notice your own attitudes, reactions and actions
it is easier to notice the mote in your own eye.
Do we really want to keep people with these skills out of the ecology of moral discussion?
But what is the point of discussing moral issues to begin with? What can we hope to achieve with them?