That sounds like real nice theoretical stuff. The reality for me is that my son does go to a private school, totally unsubsidized, and the cost of tuition is about half what is being spent per student in the Jefferson County, CO. school district [the county where my son's school is located]. While the public school students average in the 50th percentile, at my son's school, the students average in the 80th percentile. Please don't bring up the tired old argument about the public schools having to take everyone, because they do not test the special needs children.
I worked in the Information Systems department of a 50,000 student district for several year.
The the average achievement for students was dismal.
However, in what were called "Magnet" schools, schools where some type of specialized education was provided, the records of the students equaled those of students in private schools.
In this district, entry into a magnet school was "first come, first served".
Parents who wished to have their children enrolled in one of these schools would spend a day or two camped in like to apply.
The common element here is parental involvement. A parent who will sit in line or choose to pay the extra expense of private school tuition has a commitment to their child's education. This, I believe is what makes the difference.
However, it is not fair to try to compare the per student cost of the public school system to that of a private school.
This district was responsible for providing transportation for all students, not just ones attending the systems schools. This was by state law.
They were also required to maintain the health records of all students in the district.
Also, they were required to provide seating and teachers at minimum state levels within days of the opening of school or lose a portion of their state funds. They might have a good idea of attendance, based on past history, but they had to be sure to over staff because of the time it would take to get any needed extra staff online.
These are just a few of the "added" costs I remember off hand.
The major difference here is that the public school answers to state and federal regulators where the private school answers to the parents who will pay the tuition.
Parents, it seems, make the difference.