I was the average child who was suddenly thrust amongst middle to upper class children and often wondered this.
As an educator in the private sector myself for over a decade, now I can tell you why! It's not just down to mirroring their parents, though that is a factor.
If you observe them closely for long, most kids – no matter how boisterous – have a vulnerability and insecurity about them. This is because they're still learning how the world works, and they're also used to being told what to do/what not to do generally. Not in an abusive way, but most kids know adults have the final say, and that their needs are secondary to adults' priorities (e.g. on a busy day).
In contrast, many well-off kids' experiences are optional and paid for, so there's a weird power balance reversal where adults have to please the child. Not in a bratty or rude way, but I can see the difference in assumed entitlement and confidence between children who are in expensive programmes I've worked for vs run of the mill programmes I've also worked for.
Whether it's a horse riding class, a bougie art class, a hotel kids' experience, enrichment and tutoring, etc, the adult teachers/trainers/staff are there to serve the kids and keep their parents paying. It's not about the adults grovelling and simpering (though that happens in some programmes) but just showing way more deference and tolerance to each individual child's preferences, plus showing (or faking normally!) lots of appreciation of and interest in each child's opinion/self expression/etc.