There is not as much state control as you think, Yes there is curriculum, but if there was state control they would control alot more.
Any “Subsidies” for private schools are not popular as it is and any accomodation would likley be elimiated like Busing for private schools (Dont think its not on the table) or “Mandated Service” they can force you to send your kid to public school. Bloomberg essentially did this, They fought everyone who tried to get some accomodation outside public schools even though initially it costed the city more money in legal fees than it was worth. But many parents were scared or unable to pay for legal fees and gave in