There are some basic misunderstandings by the OP
Fairness does not mean giving the same to everyone - if you have a starving child and a well fed child you wouldn’t feed them identically- fairness would be to give each what they need - equality therefore is not about having the identical same but trying to get all what they need...
We don’t do well at that - but the answer is not to remove what is excellent and leave everyone with what is less good on the basis that they all have the same (labour / communist approach) - the answer is to find ways of giving the same opportunity to all... even if it means that the actual education is different for each child...
From a social perspective - Eton is one of the most socially diverse schools in the country - they are moving to a needs blind approach where any child who would benefit from an Eton education, regardless of background / social class / wealth will be able to go even if they need a free place to allow that - they have people out all around the country looking for suitable boys in school systems who wouldn’t traditionally send a child to Eton...
Many other private schools are also looking at this approach - they also tend to have more of a social mix as they offer a mixture of bursaries, scholarships, etc and take in children from a wider catchment area.
On the other hand most state schools are a very narrow social mix as the school tends to take children from a narrow area and therefore similar housing / social mixture... in fact good state schools will increase demand and narrow the area and influence house prices making it even less diversified
So the reality is very different from the left wing inaccuracies the OP is spouting - getting rid of private schools would weaken our society not make it fairer...
On the other hand - dealing with child poverty (poverty at a wealth / emotional / parenting / opportunity / etc level) would be a very good thing- it is just much harder than the easy jealousy driven target of get rid of the ‘elite’ which needs no thinking or intelligence to propagate as a concept ;)