There is some incredible insight hovering in the background here but I cannot put my finger on it.
I am am trained as a lawyer and therefore "disputes" are my livelihood. That being said, I have engaged in all sorts of "dispute" resolutions over the years from jury trials to bench trials to arbitrations to mediations to just simply getting people together and trying to work things out.
The schoolyard fight example is intriguing. We speak of many things when dealing with dispute resolution. I think, if asked, most people would claim that one of the prime goals would be justice or fairness. You are correct that the goal of the teacher is to end the fight. However, in that circumstance, he or she is merely playing "cop" as opposed to "judge." I think it is fair to say that if a fight is occurring we want our police to stop it and not really worry about blame. Likewise, if there is a genocide occurring I would think that the initial priority of our government is to stop it.
However, therein lies some of the insight. We would probably be outraged if the police suddenly started "judging" on the spot and were not trying to merely "stop" the fight but perhaps assist those in the "right" as it may be.
The issue of "judging" is therefore different from the initial policing of an event.
Interestingly enough, I think that we would not actually want the policeman to also be the judge. While we would insist on "neutrality" in responding to the fight, we would probably not be happy that the policeman also judged the "fault." You would think that this person would be in the best position. However, in fact, he or she may not be. This person may have come along at a time when the victim appeared to be the aggressor. This makes he/she now a partial witness and therefore comes to the dispute with a predisposed "bias."
This now poses the question of what kind of "judge" do we want.
What is interesting is the social status involvement. I had not really thought about it that way. While judges are (as you point out) willing to make decisions (within constraints as I will discuss in a bit), I have noticed that arbitrators (ostensibly private judges) are somewhat less willing to do that. They sometimes try to play the "wise neutral" and come down with decisions that are ambiguous. Mediators are arguably there to "mediate" and therefore should be neutral. Nevertheless, that "neutrality" seems to get in the way sometimes. The best mediators sometime resort to tricks to get agreements. They give the appearance of leaning your way in order to get you to agree.
What I can say is that getting to a decision in our system is a long and difficult task. Even in situations where both parties are eager to get a decision, any decision, at times judges may try to dodge the decision. There are, in fact, constraints on judges that are separate and distinct from those on layers or lay people. Judges are worried about being overturned by appellate courts. They have their own reputation for efficiency to consider. They also have judicial precedent to consider. These are al "other" outside factors that serve to limit their decisions or provide some other basis aside from just personal whim. (A "proxy" for neutrality?).
This is a bit of a meandering discussion. However, the insight that an individual does not want to appear to "lower" themselves to the level of the disputants is an interesting insight that bears further thought. The "false" neutrality based on consideration of social standing could be a bar to achieving optimal dispute resolution. Perhaps there are situations where attempts to achieve resolution have failed not because the involved individual were not "high" level enough but maybe they were, in fact too high up the social scale and that we can expect their reaction to be "neutral."
After all, the concept of a jury trial originates as a jury of our "peers" not our overlords or those less than us. I had always thought that this was concerned with "bias" or perhaps knowledge of similar circumstances. I thought it was a system designed to "protect" those being judged. However, maybe it is really more than a "fairness" issue. Maybe it intuitively recognizes the inefficiency of having those higher (or lower) than us make decisions? Those higher than us may be less inclined to really appreciate the importance of those decisions and thereby default to "neutrality." Those lower than us may be too eager to make decisions to establish their standing in society.