I am confused. 20 mile commute to school is feasible personally for your child but moving 20 miles to be in the district suddenly makes taking care of certain individual infeasible?
I'm not immediately aware of any studies, but I would be shocked if there wasn't a correlation. Low-income families have a higher probability of raising children in a dysfunctional home who then become a burden on the local school district. Respectable teachers then flee these schools for better opportunities. This part isn't conjecture, as I'm recounting personal testimonies of teachers who have come and gone from my town. Whether this is a widespread issue, or how much of the problem it represents, I cannot confirm.
Many American school districts are funded by property taxes, and access to higher quality schools is valuable. It follows that communities with more desirable schools would also have more desirable housing.
> Many American school districts are funded by property taxes
This is a red herring. Local taxes are not the only source of funding, and state / federal finding is deliberately allocated in a way that makes up the difference. Or iirc often more than makes up the difference.