(sent to law courts)
Two words: cognition and casuistry.
1. Judging isn't its own activity. Everything you have just asked could be asked as follows: what is good thinking? What is good cognition? If my brain is limited to the syllogism and analogy (formalistic), can I get the job done properly? (answer: no). If I just think based on my experience or my feelings, am I thinking well? (answer: not really). Let us not ask what is good judging, let us ask what is good cognition.
2. When one has to act upon cognition in order to make judgments in individual cases (based upon reasons), the issue becomes whether we are engaging in good CASUISTRY. How can I be a good casuist? (Answer: to think well and know all you can). Asking whether someone is a good judge is no different, really, than asking whether they are good parents or professors or peer-review contributors or anything that uses the brain to think. The proof is in the product.
3. The issue you are really raising here is where 1st-person experiences fit into cognition. We know that cognition has an analytic and synthetic component. We all know that part of the brain's job at "getting at the world" is to get at experiences and phenomenology. Good brains analyze as well as they synthesize. So as much as we ask what a judge's "vision" is, we should be concerned with how their brains processes things. For me, I would care less about who wins or loses so long as the product was considerate, in good faith, coherent and had integrity.