Even if nationwide absolute mediocre student body was a goal banning private schools wouldn’t achieve it.
Next you would have to ban tutoring companies, after that you would have to ban test prep, after that private tutors, after that you would need restrictions on funding for all schools (which wouldn’t work since not all schools have the exact same funding needs), you would still have advantages. One kid is closer to the library, one kid has a parent who was a teacher, one kid has a stay at home parent with the resources to help them with homework, etc.
Nothing short of an absolute police state of fairness would be able to achieve this.