Yes, Tim nice but dim might get foot in the door in a few places (but not most).
However the education many private school pupils receive is better so Bill (private school, not dim) does better than Kevin who went to a comp and even Branson who didn't do well at school was at a private school. You find children from private schools who "fail" often have other aspects of the education they can later use., In fact my children's father a teacher in both sectors used to say that in some ways the children who were not so bright tin the private schools he was in were the most benefited by the schools because they could be taught in very small classes and helped to perform to the best of their ability and work around whatever problems they have.
If you can't afford to pay for a good private school there's no point in anyone beating themselves up about it although you could help the cause of women by getting a well paid job that pays school fees (we need many more women doing that to try to right the balance with men) but do look to the private sector.
I think I mentioned a few things above like ensure the children get 2 grade 8s in music, have them running for the county, get their D of E awards, ensure they have a wide range of hobbies, get them useful work experience, ensure they know how to write proper English, spell, pick them up if they muddle it's and its, choose for them proper GCSE subjects (look at the better private schools' lists of A levels and GCSEs and ensure they do 8 core traditional GCSEs, ensure they are with children who will be going to the best universities not the local poly), teach them the national anthem, take them on trips.
What could the state sector learn from the private? I think some state schools do have similar things that seem to work already like houses and cups and high expectations. Avoiding socialist teachers is a big big issue. It's always been so. Teachers hvae always been fairly badly paid for their expertise so they tend to be left wing. YOu can avoid that to an extent in a fee paying school as the real left ones wouldn't set foot in it but it's a problem at some state schools. A left wing lowish paid teacher may be telling a child they wouldn't fit in at Oxbridge or that being a banker is a morally wicked choice.