This sounds like a good idea in theory and I hope it works in practice.
Unfortunately, it might drive the costs of private school tuition up just like government loans drove up college costs.
It might. I went to a private conservative Christian High School, they ran on a shoestring budget, my parents had to pay tuition on top of property tax for the public school that I did not attend (my tuition was about the same as what the school district received from the property tax) and pay for transportation back and forth to the school until I was old enough to drive. It was a huge sacrifice. The teachers only received pay for the months they worked, and it was around min wage.
Now the school I went to, I doubt they would allow any govt money or voucher. They are too libertarian and insisted on separation of church and state, they did not want the state up in their business, other then the necessary requirements for a school, like running water and certified teachers.
I think it will keep the schools in check due to being opened up to the free market. Most private schools (k to 12) educate children for a lot less money per student the states do.
I hope the gov't does not make a lot of stupid requirements for the private school and tutors that create extra layers of govt, but that is probably what will happen.