I attended a private high school like that, and one of my parents worked at a similar high school. The other worked in inner city schools in the NYC DOE. They encountered very different types of parents throughout their careers, yes, but I'd describe the difference as being more of a continuum of helicopter parenting, not litigiousness.
The two private schools I have some experience with (the one I attended and the one my father taught at) had no problem disciplining - and even expelling - rich kids. The only time conspicuous leniency was granted for serious offenses was when the kid was 1) a first time offender, and 2) the child of a large donor or member of the school board.
Sure, the occasional lawsuit comes up (more often than not it's just a threat of a lawsuit). But the schools I'm thinking of charges $20k annual tuition to most students; scholarships reducing that tuition are both competitive and limited. They also have a large, consistent stream of donations. They can weather the storm on most such lawsuits and have counsel on retainer for that kind of thing.
Having known people who went to private school, and discussing the differences between public and private schools, one of the biggest differences is the balance of power of the administrators and teachers. Getting kicked out of private school, especially ones that take a terms of service like approach in their legal language and contracts, means that on average students show a lot of respect to their teachers and to administrators. The private schools my friends went to all generally had a policy of having to refer to staff as sir/ma'am, had very strict codes of conduct, and generally acted quickly to kick trouble makers out of the school.
Having gone to a private school in addition to being able to easily remove trouble makers from the school they were also able to extend a hand to these same students before expulsion. At least at my school they really wanted students to succeed.
The school will kick them out if they are. Public schools have to accept all kids who live in their area, private schools can refuse to accept the kids.
I grew up in a very wealthy school district and we had a big private school in the suburb as well. The local cops were leery of disciplining kids, let alone the teachers.
I think it’s less likely the parents than the institutions. Most of the private schools are hard to get into. If the parents sue one of them, their kids are probably not getting admission into any private school.
A public school, OTOH, won’t be able to block a child’s admission on the basis of their parents’ litigiousness.
Look at it this way. Would you rather deal with parents of one troublemaker, or with complaints of multitudes of parents of other kids that have their well paid education degraded and interrupted by the troublemaker?
Value of these schools depend on how well they manage this.
