it is interesting to look at germany here. rules there are different in each region, and some regions have free choice of schools. basically the finding is that in some cases free choice lead to even more segregation in each school because parents avoid schools that have lots of lower class or migrant children. by forcing parents to live near the school the problem is "limited" to the people who can afford to move. so it's swapping one problem for another. the results seem to be mixed however, some studies report that switching to free choice had no effect.

I’m experiencing the same in New Zealand. In-zone houses for high-performing public schools are more expensive than adjacent houses (I.e. out of zone) by almost exactly the amount of the local private school fees. House price scales with the number of bedrooms in the same way that private school fees scale with the number of children enrolled. So the question is; is it more beneficial for the children to get the more expensive education or to inherit the more expensive property?

Side note, “public school” here means state-owned and “private” means “owned by private individuals”, which I have heard is the opposite nomenclature to what’s used in the UK?