Can't speak to the academic argument on state vs private beyond opinion, but I CAN speak on first hand experience with the two groups over many years.
My job (food service) means I interact with the full spectrum of class & let me tell you - there's a wild, WILD difference between state & private school kids. There is a week before holiday season kicks off when it's just the private school families that are here & it's the worst week of the whole season. The kids tend to be incredibly rude & demanding if they deign to talk to you at all, even into their teens they'll often not look at you when you talk to them & will talk through their parents. They are FAR more likely to have 'allergies', specific food requirements, need to change menus or make up dishes entirely. They leave the most amount of mess, chairs strewn around, rubbish & food on the floor. Those families are the least patient & most likely to get agressive & nasty if things go awry with the dads being the worst.
If it's not the above then it's children being ignored by the parents whilst they get drunk/treat themselves & the kids brushed off - sometimes with the nanny they've brought with them who is similarly ignored.
There's an arrogance & aloofness that state kids just do not tend to have &, frankly, the private school kids just aren't as happy in that family setting.
Prior to my 20yrs in this part of the world & this industry I'd never have imagined this to have been the case. I'd have assumed the opposite & it absolutely informed my decision to use the state system.
(for clarity, this is not just my experience, it's across the board in this industry for staff EVEN those who were privately educated themselves!)