I've yet to work at a family owned company where the children were anywhere near as competent as their parents, and I've worked at multigenerational companies, so imagine that.
The one I spent the most time with was a publisher whose founders were Tulane educated intellectuals who walked the walk and talked the talk. The next generation sounded like they fell off an alligator tour air boat with the intellect to match. Their children are the dumbest group of human beings I've ever seen graduate from high school. Utterly and completely useless for anything other than getting hammered on the weekends and keeping random desk chairs from rolling away. Yet they were guaranteed jobs, even as the company's numbers continued to dwindle and they continued to use far more resources than they contributed. No Christmas bonus? That's because they needed to pay these morons enough to live in the same neighborhood as their father, meaning that they needed to make about 80% more than anyone else in their roles.
Ownership is not leadership. Leadership takes a set of skills that many people don't possess, and it's less common with the children of the well-off.