>>The obvious counter example is the US, where public schooling is sectioned off by neighborhood and rich kids still attend school together because they can afford to live in their respective neighborhoods.
This happens in the UK too. People who are unwilling to send their children to private school will instead move to a more expensive house because it is in the catchment area of a good school. This is morally equivalent to using extra money to sending your child to a private school.
Near us (South London) the price difference between a 2-3 bedroom house in particularly good catchment area vs. an average one is at least 100,000 pounds, sometimes more...
Which makes it very clear that it's priced based on private school costs, given that the private schools near here costs ~10k/year (huge variations, but that's roughly it for 2-3 of the most popular ones)...
I've had the argument with my ex. over my sons school too. I had halfway given in and agreed to send him private, but we ended up sending him to the local school two doors down from me instead of 40-60 minutes travel to the nearest suitable private school, thankfully. But we'll be having that argument again over secondary school, I'm sure.
This happens in the UK too. People who are unwilling to send their children to private school will instead move to a more expensive house because it is in the catchment area of a good school. This is morally equivalent to using extra money to sending your child to a private school.