I'll try. In answer, some personal experiences and a few references. Apologies for length of post.
I had thought mildly about private schools over the years, but some things really came home to me in particular when one of my sons-in-law had some difficulties ascribable largely to his schooldays privately educated.
This wasn't unusual, in my experience. Over many years, at university, socially, in working life, I have come across - in many cases got to know quite well - a largeish number of privately-educated (mostly) men and boys.
Recall the time when the Prime Minister, the Mayor of London and the Archbishop of Canterbury (head of the Established Church) were all alumni of one school? (Eton, of course.) I've known lots of Etonians, been friends with some, even. Often good people to have at a party, good fun ... but not a one who could be trusted further than you could spit. That particular characteristic untrustworthiness is, in my experience, too ubiquitous to be ascribable to anything other than what Etonians share: Eton.
(We know - I don't need to detail here - the depredations wrought on the public weal by those three individuals I mentioned above.)
Harrovians are somewhat different to Etonians, it's true; Wyckehamists different again; and so on. (I find it really weird that I can often tell the difference, btw.) What they all share, in my experience, however, is a kind of emotional immaturity allied to a capacity for bluff. They are not - again in my relatively extensive but still obviously limited experience - anything like the fully-rounded human beings we would like our own children to become. (Do I want my son to grow up like Boris Johnson? -No!)
These disabilities are associated with English private school alumni. It seems not unlikely to have been engendered by those schools, particularly if you look, as I have done in some detail, at what goes on in the schools themselves.
It appears, that is, that the education provided by these schools, despite their reputation, is actually pretty bad. And their alumni, damaged human beings as they are, take (grab?) much more than their fair share of places running things in our society. The results of this canot be other than deleterious.
That's all from my own experience. Some external readings ...
1 Sad Little Men: Private Schools and the Ruin of England, by Richard Beard.
(Beard was interviewed by the Independent newspaper: “My book is a witness statement from the scene of a crime,” he says, smiling ruefully. “Having gone to a private school at exactly the same time as Cameron and Johnson did, I saw what they saw, and learnt what they learnt. I hope my book goes some way to explaining such a mindset, and why what’s happening in the country right now is happening.”)
2 Posh Boys: How English Public Schools Ruin England, by Robert Verkaik.
(Verkaik takes an in-depth look at the system: how it operates, and how it serves to keep the country in its state of imbalance.)
3 Engines of Privilege: Britain's Private School Problem, by David Kynaston and Francis Green.
(Kynaston's part of this is possibly better written than Green's. But anyway this gives the sort of detailed coverage you might expect if you've read any of Kynaston's other (rightly respected, imo) books.)
One more thing. I know most people posting here in favour of private education do so not with regard to Eton, Harrow, Westminster etc., but rather with local SEN provision and so on in mind. Yes, you have a point about such provision; it is lacking and seriously needed. For what it's worth, I back you on your demands for such provision. Do be careful, though, of being used by those who care not at all for your and your children's needs, but who might use your justified concerns to back up their own agenda of saving rich people's hold on the governance of our land (in part) by keeping this particular subsidised private education available in the light of all the evidence of its bad effects on our society.