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.
This is a dramatic oversimplification, to the point of being entirely misleading. Money =/= better schooling. Parental involvement, cultural attitudes about education, and not using a one-size-fits-all approach is what makes schooling improve. Is the implication is that if we forced rich people to send their kids to public schools, they'd use their rich person influence to raise taxes on themselves in order to get more funding?