I believe the point of Flanagan’s article is more complicated than “private schools are bad”. Here’s a quote from the article that sums up the thesis. She doesn’t despise individual private schools, she despises the fact that in our education system, individual wealth/spending has a strong correlation with student outcomes.
In a just society, there wouldn’t be a need for these expensive schools, or for private wealth to subsidize something as fundamental as an education. We wouldn’t give rich kids and a tiny number of lottery winners an outstanding education while so many poor kids attend failing schools. In a just society, an education wouldn’t be a luxury item.
She’s basically saying that on a systemic level, wealth shouldn’t be the main factor determining whether your kid gets a good education. That’s why she advocates for investment in public schools, or specific policy or school-level changes limiting the amount that individual parents can give, for example. But then on an individual level, because she has the means to pay for a quality education for her child, she’s chosen to do so.