One of the frequently heard criticisms of private schools (or the private sector in general) is that there is a lack of accountability.
Today I was attended a focus group hosted by my high school alma mater. The school wanted to know how to reach alums, how to get them involved in the school, and, let's be honest, how to get them to donate money.
Now there is absolutely no way that someone would donate money to a school if he/she didn't learn anything there or if the overall experience wasn't a positive one. As such, a private school, if it is to survive in the long run, must provide a good education.
If a private school is unable to provide a good education, or at least one that is better than the free alternatives, it will cease to exist. And you can forget about alumni donations.
Now contrast this with public schools. Public school systems do not have to hustle for private donations. They will get funded regardless of whether they offer a decent education or not.
A public school can offer a subpar education and still survive. A private school that does not offer a decent education will disappear. A private school therefore is accountable for educational results while public schools are not.