Chicago, Schools, and Segregation: A Misunderstood Solution?

Chicago is closing a bunch of schools that they allege are being underutilized. It's an emotional issue to say the least. (This video seems to be going viral.) One reason that it is so emotional is that the schools that are targeted appear to disproportionately serve poor and non-white students. I understand people's concerns about endangering students by forcing them to travel further to get to school, often through violent areas, and ideally, the students should attend quality schools as close to their homes as possible. However, the argument that a school that has been made inefficient because it was abandoned by elites should be protected seems backward to me. Couldn't closing those schools be a way to integrate de facto segregated schools? What an interesting model for change, to close any school that has a disproportionately disadvantaged student body and force those students into adjacent school districts that are disproportionately advantaged. There would be nowhere to hide for those trying to "escape" "bad" schools (e.g. white flight); they should simply be forced to improve the schools they have. In other words, the root of the problem is not that the schools are being closed but that they have been put in such a precarious situation by privileged families following a short-sighted notion of self-interest.

Am I missing an important part of this story here either theoretically or practically? Admittedly, this a complicated social issue on which I am not an expert.