There’s a big difference between “we can’t trust people to be fair” and “fairness is irrelevant”. Irrelevance means that the rules remain the same even if people do try to be fair. In the arbitrator example, it may be that the arbitrator is trying to be fair but neither family wants a fair outcome, or it may be that the arbitrator has different ideas about what’s “fair”, or the arbitrator may be a whole company/government/institution in which each individual is trying to behave fairly but there’s selection pressure for those who pull in more resources. The picture ends up similar in all cases: the arbitrator has the power to impose their own preferred solution, ignoring the will of the families, so long as they don’t push too far and break the Schelling point (i.e. so long as they don’t demand so much so fast that the families mutually agree to find a new arbitrator).
I am not saying that the couple is behaving unfairly because of their own self-interest. I am saying that whether-or-not the couple is behaving unfairly is irrelevant; the rules of the game remain the same either way, and behavior can come from selection just as easily as intent.