*I think we sometimes confuse two types of "house price schools". There are areas in cities where house prices have risen because of particular schools and they do seem to be quite socially exclusive.
This isn't quite the same as schools (often the only one in town) where prices are high because all house prices are high in the South-East. So house prices are high in say Wantage in Oxfordshire but as its a one school town it has 16% disadvantaged pupils which is actually close to the national average of 18%.*
That might be the case in one school towns. As soon as you get somewhere with 2 or more schools, one becomes the good school and house prices in its catchment increase significantly. The other becomes the failing school which no one wants to send their children to. And so it continues.
The outstanding school is then over subscribes, the catchment gets tighter, the house prices in the catchment go up.
Or, in the case of my closest school, you draw your catchment area to include the posh areas and exclude the poorer areas. It would be cheaper for me to send both my children to private school than to pay the extra to buy a house in this schools catchment.
I don't live in London or a big town.