The government should start the process by giving back to the parents ( dollar for dollar) any tax money equal to the money spent on private school tuition.
This alone would encourage the development of private schooling.
Also...The average birth rate for each woman in the U.S. is about 2.1 children. That is about 1.05 children per adult in the U.S.
If an adult citizen were to privately educate 1.5 to 2 children then they should be fully and completely exempt from any more school taxes for the rest of their lives! They should also be refunded any school taxes they every paid previously. This money could be paid to a private foundation awarding private scholarships to children attending private schools. Or...the adult citizen could privately sponsor 1.5 to 2 children that he knows personally.
1.5 to 2 children would be more than enough to cover the expenses of paying for the poor and the profoundly handicapped.
So under this proposal, a Muslim immigrant enclave could "privatize" and incorporate a local madrassas where their children would be educated and indoctrinated by muslim clerics, in thier native language. The children could be effectively insulated from any other cultural influence. The school could charge whatever they wanted, and the parents could claim tax exemptions for that amount.