All of the people talking about the private vs. public property issue have no idea what they are talking about.
The 9th Circuit is analyzing this under SCOTUS precedent from the 1960s which held that the 1866 Civil Rights Act (now known as Section 1981) bars discrimination based on race in private contracts. That decision was then later naturally applied to private schools. You may agree or disagree with that decision, but it is the precedent here and it's what the 9th Circuit should have been working under.
What the 9th Circuit did here was disgusting. It totally twisted the Court's prior holdings to justify something the Court has expressly prohibited. The system in question here does in fact discriminate entirely based on race. The defenders of the system don't even deny that.
Arguments about the quality of the schools do not matter. It was the 9th Circuit's duty to apply the law correctly and it failed to do so. I will be shocked if this decision isn't reversed.
I should note that there is no exemption from Section 1981 for the Kamehameha schools. Congress could exempt them if it so choose, but it has not done so and thus the system should have been found in violation of the law by the 9th Circuit.