Is public schooling failing in America?

The ideal of public schooling is that all children can receive a quality education regardless of their parents' income - a laudable goal.

By running and funding schools using very local school districts the US has basically removed this aspect.

i) Schools get funding based on the means of the taxpayers in their local area - enormous disparities here.
ii) People 'pay' for public education through higher house prices - even to the point that local school quality affects house prices a great deal. Want a better school? You'd better buy a more expensive house - absurd!

iii) Students within a particular socio-economic class are much more likely to study together, because they all live in the same local area. Not so much mixing.
iv) And parents in rich areas will do more to push for good education at their local school, while providing little cross-subsidy to other schools.

Looks like a glorified private schooling system to me designed to serve elites.

To serve an equalising function school funding and management zones need to include some wealthy and less wealthy area and even out differences across them (e.g. a state, or a city), as in Australia.