States already regulate private schools. They do so whether or not a state has a private-school choice program. The Council on American Private Education (CAPE) reported on Nov. 30:
The New York Board of Regents has proposed a rule that would subject religious and independent schools to “all statutory provisions, Rules of the Regents, and Regulations of the Commissioner.” The proposal would also give the State Education Department the authority to order the closure of any elementary or high school that it finds not in compliance with all statutory provisions, rules, and regulations of the commissioner. Private school advocates in New York, led by the state CAPE, have pushed back, arguing that the rule would fundamentally transform the relationship between private schools and the state.
New York does not have a private-school choice program.
State governments can try to pile on regulations whether or not school-choice programs are involved.