Not rtfd, but I really can't think of anything that a similar background wouldn't be doing in the state system.
There will be a range of backgrounds at any independent, from the traditionally working class on a bursary, first/second generation immigrant, wealthy but abusive/ neglectful parents and everything in between. As well as the stereotype of the typical background.
The one thing they all have in common is privilege. Whether that's the privilege of being able to access the bursary system through parental involvement and ability, parents able and willing to make sacrifices to access what they believe is the best education, or the privilege of class and/ or wealth.
The only thing I can think of is all those different privileges rubbing off on each other.
Anecdotal, and certainly won't apply at the type of independent that attracts large numbers of parents there for the snob value, but I'd also say at dd's the sense of security in the parents own social status is part of it, whatever that status is.
Ime the social group where the most snobbery occurs is the lower middle class. Some seem to feel a need to affirm their own social status either by putting down the working class or deprived, through inverse snobbery, or both. And certainly at dds, and experiences I've heard about other similar independents, that mentality is such a small minority they don't have the opportunity to reduce other dcs confidence.
Also won't apply to all independents, but the pastoral care. State schools with all the will in the world simply don't have the resources to support every major problem that could destroy a child's confidence, let alone minor temporary dents. Whereas an independent that values pastoral care does have the resources