I think the implication is that if people with means to act with authority, power, and know-how on the day-to-day operations of the school actual sent their kid

I think the implication is that if people with means to act with authority, power, and know-how on the day-to-day operations of the school actual sent their kids to the schools that all the kids go to (even the kids that don't have parents with authority or power or know-how), the schools that all kids go to would be just as good. Wealth is a clumsy proxy. A single-income home, with a parent deeply invested and active in the child's education (rather than how to get food on the table next week), implies a lot of unmeasured but real resources. Resources that are more readily invested in "communities with means." Leaving the communities without those same means deprived, deprived of resources that are not easily tracked by "how much money does the school get." Of course wealth can play a role too. Consider a poor neighborhood where some of the money from the school goes to paying for school lunches, where in the wealthier neighborhood students are typically sent with their own lunchbox. It's not just about the money. It's about the people that have the money. It's not something you can easily policy away, imho, because it's a systemic cultural issue, with a lot of messy tangles.