The Toff always chats amiably to the Taxi Driver. But he most definitely doesn't want his children to attend the same school as the taxi driver's. Fortunately, Eton exists.
The lecturer at Middle England University is always polite to his or her house cleaner but, of course, their children go to different schools. Sometimes socially segregated housing does the trick automatically; but when it doesn't, God steps in. Faith Schools boomed under the encouragement of Middle England New Labour and continue to boom under their successors. And so nowadays lecturers at Middle England University Do God. It reduces cognitive dissonance for them to believe that, actually, God isn't so bad after all. He improves your children's GCSE results.
The Churches are well aware that this is a bigger selling point than Eternal Salvation.
At the bottom of the heap, the children no one wants to love go to common schools.
Once upon a time, the common school was regarded as a vital institution of a democratic society. It was what taught children (and their parents) that they were all in it together, that you could only progress where there was some social solidarity. If there was to be any differentiation, it could only be in order to give talent a challenge or lack of ability some compensating help. There was no place for differentiation on the basis of social background.
Some countries seem to maintain traditions of common schooling. But not England. We do of course believe that we are all in it together: the rich man's children remain in his castle and the poor man's children at his gate.