It's an interesting conversation, and one that has come up on MN for years.
It's definitely true that not all private schools are equal, and not all state schools are equal - so it's probably impossible to draw any definitive conclusions.
I come from a family of 6 siblings, all of us state educated. We all grew up in Scotland, so state comprehensive schooling was the norm for most folk.
One of my DB's has done exceptionally well for himself, a high earner in London, as is DSiL.
My niece and nephews are roughly the same age as my 3 DCs.
DB put his 3 DCs through private school, at a cost that was eye-watering to me.
I put my 3 DCs through state school in Yorkshire - a good local school, and we didn't need to pay a house price premium.
All 3 of my DCs achieved higher academically than my brother's DCs - and my sister's DCs also achieved higher grades.
However - we all attended a family event last summer, and it was quite clear to me that my niece and nephews had an air of confidence, that my DC's didn't.
My niece and nephews had the ability to join any conversation with anyone, and were fully able to offer their opinions on any subject.
My DC's tended to hold back? They were polite when spoken to, but seemed to lack the confidence to enter themselves into conversations?
My niece and nephews are all still living at home (in Surrey), and have secured themselves professional jobs - lined up by contacts of DB and DSiL. They are earning very well, and will soon be able to afford a deposit on a London property.
My DS, with his better degree grade, is working hard, paying his rent in Newcastle. He doesn't rely upon me financially whatsoever, but is highly unlikely to be able to save up a deposit.
I don't have 'contacts', and DS doesn't have 'networks', and he definitely isn't as "polished" as his privately educated cousins.