If you are ideologically opposed to private schools full stop and would ban them tomorrow, how do you think that would help the state school system and help poorer children in it, when there would be exactly the same amount of money in the education budget?
Schools would need to absorb the 7% of children who currently have the 'pushy middle class parents' who privately educate them. Do you not think they will use their money and their ambition to make sure they monopolise the best schools in the best areas? There are roughly half a million British kids in UK private schools right now. That's the equivalent of 500 new schools each accommodating 1000 pupils that will be needed, if everyone woke up tomorrow and decided they'd grant you your wish for equity and fairness. Maybe a few less after you've forced people to take places in the failing, undersubscribed schools, except that Ottilie and Orlando won't be going to one of those, will they? Their parents will be buying a house in the leafy catchment of a super-selective grammar.
Where is the money for that coming from? What people like you fail to see is that whether it gives an advantage to children of well off parents or not, the existence of the private school system in this country actually does the department of education an enormous favour and saves it an absolute fucking fortune.