I think this agenda is a little more complex than you seem to suggest. That doesn't mean we shouldn't do so--but real improvement will have to be much more systematic and revolutionary than some simple restructuring at the local levels.
Segregation is the symptom of the problem, not the root of the problem. Give people in power the opportunity, and they'll game the system in order to ensure that their kids have an advantage over everyone else.
I'm not saying that the students or teachers at these schools are in any way inferior. But from what I've read, these schools are underfunded and ignored by the education administration (Betsy deVos once visited one such school, only to badmouth it later for what she considered defects. During a tour of NYC schools, she skipped public schools and visited wealthy private religious schools.)
Discrimination based on race and class is so endemic in our public education system, that small improvements will only reinforce the inequities.
Nor is systemic racial / class discrimination limited to public education. Compare the endowment of Howard or Morehouse or Spelman to that of Harvard or Yale.
Charter schools are not a positive alternative either. Students who attend either publically or privately managed charter K-12 schools may perform slightly better on standardized test scores than students at disadvantaged public schools, but they are still far behind students who attend wealthy private K-12 schools in terms of achievement and opportunity.
So in the long run, it doesn't matter where people send their kids or which local petitions they sign. If we're not committed to ending systemic racial and economic discrimination in the ENTIRE education system, we're not part of the problem, we ARE the problem.