Tbh, I wouldn't dream of telling my dc that privately educated kids are all entitled, arrogant and stuck up. I don't think that's true in any case, but even if I did, I would leave her to make her own assessment. And I certainly wouldn't be slagging off kids who are her friends!
As it happens, she did previously form a rather negative view of the boys at one particular private school, based on her perception that they were arrogant and entitled. That perception did not extend to any of the other private schools in our area, however. It might surprise you to know that, far from reinforcing that perception, I was at great pains to point out to her that a few arrogant, entitled twats from one school didn't necessarily mean that everyone from that school was entitled and arrogant. She does understand this, of course, and has developed a much more nuanced understanding as she has got older. She retains a very negative view of that particular school culture, though, while acknowledging that there will be some pupils who manage not to be tainted by it.
I have made it clear above that I think privately educated kids are less likely to have a good awareness and understanding of socioeconomic diversity than kids who go through the state system, and I do think that it is bad for society as a whole that so many privately educated people end up in positions of political power for precisely this reason. However, that doesn't mean that I think privately educated people are all horrible! As I have said several times, I have plenty of friends who were privately educated, and plenty who choose to privately educate their kids. They wouldn't be my friends if I hated them.
Of course, I have met some privately educated people who absolutely are stuck up, arrogant and entitled twats. I have also observed quite a few I recent years running our government. However, I have met plenty of state educated twats over the years as well. Sadly, neither sector has a monopoly on this.