I think it's a feedback loop.

It's a loop. In many areas, schools are funded by property taxes, so higher property value leads to more funding for the school, which leads to perception of better school, which drives more people to want to live there, which produces higher property values, which...

That doesn't fully explain it though. Many of the worst schools also have some of the most funding per student.

Possibly, but in any case the problem isn't that public schools are flawed as a concept, but rather that some of USA public schools are bad. If many of these schools run on the same set of general principles and processes (although often with more funding) but work, the problem is not with the concept or goal but rather with the particular (some) implementations, some of the managers, or (mis)allocation of resources.

And they moved to where there are many other upper middle class families who are very involved in their children's education. As a result, you see schools with good metrics on standardized tests and the like even though the per pupil spend is not uncommonly less than that in urban schools with poorer metrics.

Public school funding in the US is probably the most inequitable in the developed world. Not only does each state decide its funding and taxes, but smaller local areas do too. High home values => high property taxes => more school funding.