The concept of a school wouldn't be illegal. The people running the school would be committing crimes or at least be in some level of legal trouble with the city council, state or federal law. They could either be fined or go to prison, or they could get the relevant paperwork sorted out.

> The concept of a school wouldn't be illegal.

Yes it is, you explicitly said it would be illegal if you didn't get permission from the government. What you meant to say was that "The concept of a government-approved school wouldn't be illegal." which is very different from the concept of any school being legal, and also redundant because by definition a government-approved school is a legal school, since governments hold a monopoly on the legal system.