Read it all. Neither private nor church schools or for that matter home schools are free to do as they please in Alabama.
You state:
"I don't know how things are structured in the UK, but in the US, at least, schools that are privately owned and run are not subject to any kinds or monitoring or testing or any kind of government intervention at all."

Maybe you have the US confused with Canada.

There are government requirements in Alabama for church schools:


Recordkeeping/Reports: The principal teacher of private and church schools must keep an attendance register showing the enrollment of the school and every absence of each enrolled child from school for a half day or more. Ala. Code § 16-28-8. The registry is admissible as evidence in compulsory attendance hearings. Ala. Code of Ala. 1975 § 16-28-23.

At church schools, enrollment and attendance must be reported to the local public school superintendent by the parent or guardian on a form provided by the superintendent. The administrator of the church school countersigns the enrollment form. If a child leaves the church school, the church school will notify the local public school superintendent, by prior consent of the parent or guardian. Ala. Code § 16-28-7.

Transportation: A license tax or registration fee of $13.00 is imposed on motor buses owned by a church or a private school that are used only for the purposes of the institution. Ala. Code § 40-12-246 (d).

Home Schooling is also regulated:
Home Schooling: Every child between the ages of 7 and 16 are required to attend a public school, private school, church school, or be instructed by a private tutor certified by the state of Alabama for the entire length of the school term in every scholastic subject. Parents who want to have their child instructed at home must either do so through a certified tutor or must qualify as a church school. Church schools offer instruction in grades K-12 or any combination thereof; are operated as a ministry of a local church or group of churches, denomination, and/or association of churches on a nonprofit basis; and do not receive any federal or state funding. Ala. Code §§ 16-28-1; 16-28-3. The enrollment and attendance of a child in a church school must be filed with the local public school superintendent by the parent or guardian on a form provided. Ala. Code § 16-28-7.