Its very hard to prove that, And you have to go to an a beaurocrat who thinks religous schools are nonsense and when they refuse you have to convince a arbritor that private schools are better than public schools and again many of them do not belive in relgious schools and certainly do not believe in segregated schools (in the relgious sense)
That’s precisely why frum communities have voted in (legally and democratically) boards which understand the needs of the children from the community. There are a lot of issues which are subject to legitimate interpretation, and each side wants their interpretation to be implemented.