"reduce inequalities based on social systems".
Actually, I vehemently disagree that there should be a distinction between natural inequalities and societally created inequalities. Alot of "societal inequalities" come about as solutions to natural inequalities. As humans, we very much can and should have agency.
I disagree that because nature made me short, ugly and bad at maths, I should be forever deprived of all the good things that tall, beautiful, clever people enjoy. I want loads of sex too!
I know the tabloids focus on the super wealthy in private schools. Alot of these people have large inherited wealth, so private vs public schooling for them is neither here nor there.
My personal experience of state vs private - I have experienced both - is that alot of the private parents are not in fact uber wealthy (yes, there are definitely some!) but are people who, for reasons of their own/the child they have, chose - at great cost to themselves - to pay twice for their childs education. (agreed, these parents still definitely earn above average).
In the state school I have experience of (affluent area), many parents had similar levels of wealth to the private parents, but chose state. Alot more were sahm's vs the private school. Some would rather give their kid a house deposit. Some thought a nicer quality of life for their child was more important to them.
Both are legitimate choices imo. Both actually cause different societal inequalities. Do we ban stay at home mums because their kids are more advantaged (help with homework, more involved in school life, more time to be emotionally available) than the kids whose mums cannot afford to stay at home? (tbf mumsnet threads often do demonise sahm's).
Surely we should all be allowed to make individual choices based on who we are, and our strengths, preferences and capabilities?
A brush-stroke generalisation about private education, like all generalisations is lazy and unfair.