When I last looked it up, the US data indicated that 91% of custody cases are decided without recourse to the family court system. Of the 9% that go to family court system, 5% are decided after a custody evaluation, and the remaining 4% go to trial. Only 11% of custody cases go to mediation.

It appears that the majority of custody decisions are made by the parents themselves, without recourse to the court, or even to an external mediator. If the judiciary is causing a disparity in custody outcomes, it must be indirectly.