The argument is not for funding private providers. The argument is funding the child. Not really sure how you folks don't get this. For the sake of example, each kid gets $1k/month for education. How it gets spent is on them/parents. Why can't the parents make the choice? What's your argument?
"Us folks" completely understand your argument, we just don't agree with it.
As I have tried to explain, government funded services are not redeemable in cash. If my neighborhood doesn't like the way the City plows the snow on our street, we can't get a voucher from the city to hire a private company to do it. I don't get to decide that I'd rather beef up my own fire and security system by asking for my share of the cash that the City spends on Police and Fire departments.
The reason that is the way it works, is because it's more efficient for the government to plan, organize, construct and staff the departments of streets, fire and police, than it is for each member of the community to figure out how to get those services on their own.
In addition, a huge amount of the cost of public schools and City departments are fixed costs. Buildings and equipment are a big portion of the monthly "cost". The remaining costs are "semi-variable", meaning that the costs don't go down when one student or 1 customer for City services decides on an alternative.
A few years ago CR School district's enrollment was close to 17,000 students attending 31 different schools. If 3 students from each public school wanted to transfer to a private school, the public school costs would hardly change. So if the funding for those kids ($93,000/mo in your example) that the public school will lose simply cuts into the total funding for the school without a corresponding reduction in their costs.
BTW, if the government starts funding for "each child", then do I get some of my property tax money back since I no longer have kids in school?