I think the thing that makes the biggest difference is interested parents not the money spent or not spent on fees.
Some parents know nothing about education and aren't willing to put any leg work in and think that throwing fees at the issue must result in a good outcome. They think all fee paying schools must somehow be better and do t put the legwork into investigating schools properly of either the state or independent type but choose based on rather spurious reasons . YOud be surprised how many choose a Prep partly based on uniform.
When parents are genuinely informed and interested in education I really do think they can make it work. Some of those parents will choose a great independent for their kids - it will be chosen because it suits their children and provides things which will benefit their particular children, not just because it is fee paying or offers zillions of activities that their child will never do.
And the similarly educated but slightly less affluent parent can do equally well for their child. This parent has probably investigated independents and realised they can't afford them, but has also investigated the local schools well in advance and is making a plan for education. Yes, they sometimes move house for a particular school or they find out the admissions criteria far enough in advance to meet them and they then meet them. Their children go to extra curricular activity not becaue they are making up for not going to Independnet school but because those parents genuinely value music and sport at a high level and other activities. Those children might be playing an instrument and climbing and attending a Coding holiday course and going on a ski trip and going to the library.....the parents are putting in the time and effort to find out what is available and organising it - some would say that is far more valuable in terms of input than paying a set if school fees and then leaving the provision to the mercy of the school - yes the school might provide lots, but it won't be quite the same as an absolutely tailor made extra curricular programme that a parent who has the time and inclination can provide. Often for a really top notch experience, you need to be involved out of school anyway - so county sport, the national youth orchestra etc are all out of school things and although here are lots of independently educated children in those, state school children whose parents have given them opportunities outside of school are there too. And when a parent has been involved in helping their child look at A Level courses and choosing universities,mrathrather than abdicating responsibility to the school because they have paid for that service, the child will always get a more personalised and invested experience than a school can give. A competent, educated and invested parent can gather expert information that they might be lacking and absolutely tailor it. Nit doesn't have to be inferior to what might be gained in an independent and could often be bettering. It's about who the parents are and their values I think at the end of the day.
And what about the child of such parents who does find themselves in the disappointing school and who cannot go elsewhere? Even then the interested and involved parent can make a big difference. They can pursue the school about specific issues, be informed about the importance of GCSE and A Level choices and provide tutoring if needed, plus can probably provide at least some support themselves. And they can still be providing the extra curricular opportunities and helping their child develop a work ethic that is so important and money often can't buy. So it might not be such an easy ride, but the children from those backgrounds can reach good Unis too. Their expectations, wider experience and values can make a vast amount of difference. I would say that child is better off and more prepared for life by far than the unusual, very poor child from a disinterested family who somehow gets a full bursary to a private school but who still has parents who don't know about education or actively support their child or provide an environment of high expectations. Culture at home counts for an awful lot and I think people over-value the role of paying fees and forget the impact they themselves have.
Sadly, I think it's the children from the disinterested backgrounds that we need to worry about, not the stretched middle classes who can't afford fees and who worry their children might be missing out on something. Those children have all the resources needed to be successful if their parents put their minds to it.