@GodessOfThunder - You could get rid of private schools tomorrow. Some people will always be wealthier than others and the privileges this entails will just manifest in other ways. This is obvious.
The fact is, state education in the U.K. is a postcode lottery. Those with the means, can move into the prestigious catchment areas - often the cost of property / stamp duty costs more than school fees.
Grammar schools are also a postcode lottery.
Who is more 'privileged?' A child who achieves top grades in a state school in a safe area in the Home Counties with two university-educated parents. Or a child from inner London whose parents decided to pay for school to avoid their local comp due to stabbings?
Parental education carries more weight and 'social capital' than school sector. You see it all the time on the Oxbridge chats - posters making a point of saying their DC got in from an 'ordinary comp', only to later drop in that they themselves are Oxbridge educated!
I assume you are not from London @GodessOfThunder because, if you were, you would know that the majority in the very top London independents are from immigrant / non-white British backgrounds. Who is more 'privileged' in societal terms - a non-white child at a London private school, or a white child from an expensive part of Surrey or similar, ie. the catchment for a school with a catchment of pupils very similar to her?
Not all comprehensives are diverse. In white, middle class areas they will reflect that. It totally depends on location and catchment.
As for boarding schools, I would argue the privilege of these is debatable. They may suit a certain type of child. But I'm sure the majority of children, if asked, would be terrified at the prospect of living away from home at the age of 13, 11 or even 7. I'm sure you've heard of the emotional trauma reported by people who describe themselves as 'boarding school survivors.' Is this a privilege? Don't most kids just want to go home at 4pm to their own space and family? These children may come across as more 'confident' because they have had to be! No choice.
Anyway, once they graduate who cares where they went to school?