Reluctantly I agree with posters who have said that private schools are exclusive. They get their results by only working with academic high flyers. Some parents probably don't realise how high and unforgiving the standard has become. Ditto with the 11 plus exams to select entrants to grammar school.
To widen the discussion somewhat, I think the whole programme of reforms to GCSE, started in 2014, has been about introducing 11plus and grammar schools by the back door. By abolishing levels at secondary and the numerical system at primary, the government has forced state schools to seek out their own measure of attainment. Many have looked to CATs . I see it that selective independent schools have always tested, and now they are using the very scientifically calibrated standardised tests and referring to the results of just these tests when discussing progress with parents, rather than referencing more holistic measures and assessing written work as they probably did before. The natural outcome of these tests being used in state schools in Year 7 is that teaching will be targeted at helping the particular pupil to achieve their predicted grade. So no expectation of a higher grade will be entertained, and so no understanding of stretching and presenting the teaching differently will be explored. We are not so very far away from describing 11 year olds as academic or not; suited to a profession, or better off training for a trade. The notion of overcoming barriers to learning will be falling from favour, and education described only as presenting a block of learning and children being bright enough to absorb and master it or not.
I think it is true that many private schools can only work with pupils who can keep up with the pace. They are aiming to teach a higher level of ideas and foster abstract and sophisticated understanding + academic thinking - hence why they recruit Oxbridge 'teachers' to relate to pupils on that elevated level.
From what I've glimpsed of friends, private schools rely heavily upon setting mountains of homework and relying on the parents to nurture and prompt the kids to do it, for hours at a time. I'm not sure there can be spirited discussion in the staff room about ways to get the children to grasp the learning. There is a competitive ethos which makes the kids want to try their very best, however full or inspiring is the content being learnt. Home background, academic trips to museums and debate incalcate a useful level of expectation in the child to anticipate what topics and ideas they might be about to encounter - and the discussion skills to talk about these.
Finding an independent school which accepts pupils with SEN is a good route. The teaching will be stimulating and academic and suited to developing the students' skills to match the higher level GCSE and A'level grades, but the way learning is taught will be as important as what is taught.
For me the old education system did well, and barriers to learning were genuinely removed.