Given the realities that alternates to the public school system exist1, that students are not "assembled at gunpoint"2, and that the curriculum in the public school is as responsive to external pressure as those that exist in a "pay as you go" environment3 the degree of compulsion is likely equal as far as students are concerned regardless of venue (public, private, home, or work placement).
Do you intend to suggest that compulsory attendance statutes, educational neglect laws, and tax subsidization have no impact?
1. "Alternatives" exist for those who can afford them. NCES reported that support for vouchers is higher among black parents than among white parents and that support for vouchers is negatively related to family income. Poor people want out of the NEA/AFT/AFSCME cartel's wretched schools.
2. A law is a threat by a government to kidnap (arrest), assault (subdue), and forcibly infect with HIV (imprison) someone. Fail to enroll your kids and the State will take them from you. Resist effectively and the police will shoot you. Yes, students are assembled at gunpoint.
3. Flat false. Compare the uniform curricula of the Common Core with the variety of ethnic restaurants available in any large city. I recommend Chubb and Moe, Politics, Markets, and America's Schools, and Diane Ravitch,
Left Back, A Century of Failed School Reforms. It's much easier to escape a poor performing institution in a competitive market than to reform an entrenched, rule-bound State-monopoly enterprise.
Firms in a competitive market must operate
within the law. Rule-bound State (government, generally) bureaucracies must operate
according to the law, and internal rules have the force of law.