I'm not sure if you really believe that, or if you are kidding yourself.
The families closest to the school are the richest ones!
In London, the average premium to live in catchment of a top school is £173,000.
www.standard.co.uk/lifestyle/london-life/245-thats-the-postcode-premium-for-living-in-catchment-area-of-top-london-school-8953893.html
Here's an example of a so-called state school which is run (rather haughtily IME) for the rich:
www.gordons.surrey.sch.uk/_files/PDF%20Files/Admissions/073FE4CAA217AA415A74B81DF5273027.pdf
Firstly, it costs £6,483/year for so-called 'day boarding' (i.e. not boarding at all, you go home every evening), you need to pay that to go there at all.
Secondly there are 100 places each year, 32 full-boarding allocated rather opaquely according to 'need', and the other 68 based on straight line distance, which is under half-a-mile.
If you live a couple of miles away, you can forget it. If you want to move, you better have £££ for stamp duty, removals, bloodsucking EA fees, and the higher cost of the house, which again excludes the poor.
And of course if you live under a mile away but don't get in because it wasn't close enough, they'll allocate you a failing school five miles away.