School performance is not much affected by budget. The terrible high schools in DC have really high budgets. The problem lies elsewhere.

It's really simple. Kids tend to be like their parents. Parents who raise kids well have a strong preference to be among their own kind. The parent who sits down at night to check homework is nothing like the parent who interrupts homework in favor of entertainment. The parent who enforces discipline is nothing like the parent who lets kids run in gangs.

Families are in a bidding war to be with other families that prioritize education. Families without those values will bid lower, and thus be excluded.