Having done a similar analysis myself, here is what I found. Schools in wealthier areas have better results primarily because the support system at home is much stronger. I.e. A wealthy family's kids do better in school no matter where they go. Another factor, mostly in elementary school, is that wealthier areas get noticeably more volunteer activity from parents.
Many of the higher end schools in the area here (Bay Area) require volunteer fundraising, bake sales, volunteer time contribution, etc. precisely because property taxes aren’t enough and they get less than the other schools.
And the parents do it because they can (they have the ability to get the time off), and because they know it’s important in many ways.
State & local funding is less important, by far.