Some time ago I mentored a child in our local, then badly failing, comp. She was a very bright child from a large family whose parents had never learnt English, nor been employed.
She was limited to only 7 subjects at GCSE, and the most stretching GCSE math syllabus was not taught. (At the time the school had 93% on free school dinners.) She and her lovely group of friends, most of whom faced challenging home conditions, would have breezed through an 11+ which took the top 25%, and would have thrived in a more academic environment. But the day they entered the school their chances of, say, reading medicine, were lost.
I was told by her Head of Year was that my volunteer role was to persuade her that she should go to University. If she had been in an environment where her some of peers' parents had been to University this would not have been an issue.
Tiffin and other grammars would have already been out of her reach as she could not have afforded the train fare, even if she had known about the exam and had been able to arrange her own tutoring. Better schools tend to be in leafy suburbs. What we might need is a selective school in the inner city which allows bright kids from poor backgrounds to succeed. Talking about it at University level is too late. The ambition and aspiration need to be cultivated, and opportunity, needs to be there at 11.