If a county, or a state, wants to send money to poorer areas, they can do it without regard to the organization of the school districts. Often, the formula is targeted at the student: if a school has X low-income students, then they get Y dollars.
Edit: my comment pertains to large school districts, aka city-wide or within large counties. The suburbs get organized differently.
Drive your kid across town, when you’re already poor?
Under choice fewer people would be denied an education by your definition.
It is pretty straightforward economic self interest. I can recommend Sowell's book "Charter schools and their enemies" it lays out the setup quite well.
Did I send my oldest kid to a charter school? Yeah, for one year. It was a bunch of Teach for America grads that warehoused the kids in front of computers for hours at a time, no PE, and sent kindergartens to detention. (Wtf?!)
So why did we choose this school? One reason: All day kindergarten. We had no idea how bad the school was actually run until like a quarter or two into it.
Fuck Rocketship Schools.
There are downsides, small population = fewer extracurricular activities, with less participation (esp. sports).
Everyone wants the best for themselves and their kid. That means buying housing in the best district they can afford. Once they're in they don't want lower socioeconomic status people to move into their neighborhood or the district to annex poor areas.