Second, you can avoid hiring special-ed non-teachers, who (IIRC) are quite numerous in some public school systems (like 1/3 the staff?) but aren't usually counted in the teacher/pupil ratio (because their jobs are different).
Baltimore: "For every student enrolled in City Schools, $1630 goes to administrators" [1]
Even universities have a problem with the administration growing wihtout bounds.
A profit motive can cut down on unneeded administrators. The "misc_costs" can be widly different for different school systems.
Why should a head of a public school each $+100k ? [2] What is it about a standard run-of-the-mill public school that can only be performed by such high earners?
1) https://foxbaltimore.com/news/project-baltimore/baltimore-ci...
2) https://www.glassdoor.com/Salaries/elementary-school-princip...
If you can select only high performing students with fewer individual needs, you can increase the number of students without needing additional staff. Likewise, you can higher fewer or no language teachers, special education teachers, etc.