As far as I understand it, in non-Montessori schools, a lot of teacher effort is spent on the wrong things, like grading, anti-cheating measures, very strict lesson orders, and so on.

Somehow, these private schools offer better education for less money than that.

We need to stop funding schools and start funding children.

Let them take that money to the place where they learn the best. I’d bet teachers would be paid better and bad teachers would be removed much more quickly. I’d also conjecture that teachers would enjoy their jobs much more

They don't have as many "problem" kids who get one-on-one dedicated personnel, self-contained classrooms with two staff just for ten kids, et c. Nor do their regular classroom teachers have to put up with as much disruption, both because there's less to begin with and because they can kick kids out much more easily than public schools can.

And even "private schools" is too broad a category. Most private schools are no better, or even worse, than nearby public schools. Often this is because their appeal is religious, first, over education. However, many of the secular or more gently-religious ones that look better on paper may actually deliver worse education, but look superior purely due to selection bias—this can be revealed if their students move to public schools. They don't always come in and breeze through because they were ahead (to be clear, that does sometimes happen if the school was good!), instead they may be badly behind, despite having had good grades at their expensive private school.

Those numbers don't include other Full Time Employees like aids, tutors, and other support staff that work with students.