Yes rich kids can experience mental illness and drug problems. No they aren't responsible for the actions of their parents. Yes they deserve sympathy and treatment for these problems.
The issue is that...most kids aren't rich, and rich kids are much more likely to have the resources to get all of these things. They're also more likely to have people write sympathetic op-eds about them and avoid getting shoved into the criminal justice system at an early age.
The point isn't that rich kids don't deserve sympathy for their real problems, the point is our society gives them much more of it than it does anybody else.