The core problem is that the effective route to massive improvements in equality of access to quality public education is basically "solve poverty, and concentrations of poverty", which isn't within the scope of schools' mission and is going to be far harder to sell, politically, take a long time, and be incredibly expensive.
We tried "bussing", which directly confronted parts of this, and it fucking worked, which makes it practically unique among attempts to address this problem, and it was relatively cheap compared to other things we might try that could work, but literally everyone hated it (yes, even the poorer families) so I guess that's out.