That type of "networking" will always exist because it is based on background.
Those kids grew up in a wealthy bubble together, which is also why they tend to stick together when older.
Increasing VAT on private schools will in fact make that worse because it will push away the middle class from that rarified circle, further concentrating it.
If you want to fix State Schools, you first need to be objective about what the problem is. It is not kids that go to private schools, but rather it is a lack of funding (for buildings and teachers).
Increasing VAT on private schools is small potatoes at circa £1bn Won't make a dent in that.
So while I do agree there are avenues for raising more tax on the wealthy, this will only work if everybody pays more in as we have an ageing population, and far too narrow a working population tax base.
So the people complaining on here about people paying for private school, you will have to pay more in tax as well if you want the situation to improve.
The days of being heavily subsidised by higher earners for school, NHS, and various other benefits are effectively over due to the deterioraring nature of UK demographics.
"Somebody else" is not going to be paying for the situation to improve. You need skin in the game as well.