The process beings with having private schools for people to choose.
Who are you proposing should begin this process, the government?
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.