I'm not convinced OP is serious, but here goes.
I'm of a working-class, state comp background, but my education and life-style would make people identify me as middle-class.
My children are at a 'real' comprehensive school, with a large proportion of doctors, consultants and uni lecturer's kids as well as trades people, blue-collar workers and benefit supported families. This is what comprehensive should mean - all inclusive.
However my children are very definitely middle class, ( due to their life-style and privileges it would be laughable to suggest anything else.) While private school is less likely for those from a working class background, ( because of all the usual inequalities), the state system, despite what some people choose to believe, is full of children who are, and will remain part of the middle-classes.
These children also have the advantage, in my view, of gaining insights into the lives of children from other socio-economic backgrounds, very helpful if you want to reduce divisions in society. Why would this give children a 'chip'? If anything my children feel their academic successes have been achieved in a fairer environment, ( possibly tougher) and if anything this gives them the edge over their privately educated peers.