FWIW, that is how it works here in British Columbia right now in Canada. The teacher unions hate it, but for now we get 50% of the annual cost back to private institutions. Don’t know the exact amounts but let’s say the province hands over $5000 per child. The private school generally would charge $5000 for tuition from the parents, getting an equal amount from the province. Of course the private school can be elite and charge $20,000 to the parents but it still will only get $5000 from the province.
It works well, but opponents claim it only benefits the rich, which is total BS. 80% of those in our school are middle class.
Perhaps the biggest beneficiaries of a voucher system like that in America are the people of color trapped in inner city failed schools.