What r7 is mostly true... .though it's not exactly skid row, there are still a few special care schools (from disability to addiction and psychiatric or undiagnosed behavioral issues. There a couple military schools that are famed for taking young disciplinary problem child stars.) that are truly the last resort but they're far more expensive, the waitlists are longer and pretty damn isolated.
For normies - yeah, their kid is a kid is a misfit, usually a theatre/music/arts geeks or pre-homo and needed to escape their hometown before they the inevitable suicide or murder.
then you have the stage parents, socialites, schmoozers that are more interested in getting to know the other parents...
the international and expats, which it probably wasn't their first choice...
the majority of boarding schools (which are estimated around 400) in the U.S. are of the religious variety,
next up are traditional boarding schools - very few are old money, most aren't that different from public schools except they offer housing.
then alternative programs - which usually break from the public structured format, some may be more educationally intensive, focus more on electives or enrichment programs and otherwise try to pack in more than the standard school would.
then you have the specialty education programs (arts, sciences, non-traditional/specialty athletics, etc), prep schools (usually geared towards the ivys and many do start in elementary these days), military, specific disability, specific cultures or non-english, and so on... the numbers dwindle as requirements, tuition and waitlists climb.
and many more industry-related folks that can't drag their kids along on every gig and need a place to house their brats and generally feel more comfortable if their kids are around others with fam in the industry, which they feel gives them a greater sense of normality and the ability to make genuine friends.
the majority of graduates don't go in the arts. . . but obviously, may have a slight advantage if they do.