@KingsHeath53
I think you have made the mistake of assuming that your reasons for going private (a 'better' peer group, less SEN) apply universally across state/private sectors.
One of the tools that private schools use to maintain that behaviour and environment is focusing on their specialism. For some that is academics, some sport, some some music (like Cheethams). For some that lower needs SEN, some that is very high needs SEN. The school set up is then tailored to this and they can be clear about when a school is appropriate or not for a particular child.
Certainly in the expensive area I live in, there are very few high needs SEN kids in the state system. Because the parents/teachers know how to use the system and can afford legal fees.
My childs peer group hasn't changed following the move to private. The state school parents were motivated and engaged in the system and raised lots of money for the school. The teaching staff were able to communicate to parents when they were struggling to meet a childs need and parents acted on it. These were the parents willing to pay the premium for the catchment. My children were very privileged when they were in the state system, just as they are now.
I mostly moved my children to private for childcare reasons, so I could keep my job. Due to moving school locations/inadequate wrap around care in junior meant the logistics no longer worked.
The one thing I think that is measurable on your list, is smaller class sizes and this definitely support better behaviour, which is turn sports faster learning. And I would add freedom from government interference as the other luxury (very happy my kids won't do SATs). These could be used to justify VAT.
Based on what you wrote, I assume you would also be happy to see personal tuition have VAT on it too, which is a consistent argument.
I can disagree but that is a clear argument which doesn't rely on stereotyping about each sector.
State schools definitely select on parental with/income via catchment as do private schools. Both sectors have children with SEN in, but allocated in different ways across the sector. Levels of extra activities/costs vary across both sectors.
It sounds like you have found a school that you feel is a good fit for your children which does make them privileged. But don't assume your reasons/experiences are the only ones out there.