Sadly (and really only generally speaking - there are exceptions), private schools' quality is driven by market forces whereas public school policies are driven by politics. School officials obtain and maintain decision making positions and power by there connections. There is little to nothing even a group of parents can do to address this. When they do, it gets taken away.
For example, in my city, parents organize "booster clubs" to raise money for their local schools and improve the quality. But parents in poorer sections of the city are often genuinely unable to do this. For example, they have a disproportionate number of families with a single parent who barely makes ends meet and works too many hours to have time to invest in a booster club. Since this is unfair, the school system is working to take money from the booster clubs to distribute to the poorer areas. So, the parents have the incentive removed and, disheartened, give up. The school system has decided, essentially, "If those schools are going to fail, it's only fair that all schools fail."
The parents can't do anything to fix their public school, so the ones who can afford it take their kids out and put them in private schools. Ms. Benedikt is correct that there are Bad Persons at play. She is dead wrong about who those Bad Persons are.