It's easy to assume that wealth and privilege make up for emotional deprivation, and of course, they don't.
The Royal Family has been, is, and always will be, deeply dysfunctional, a combination of who they are as people, and what they are as an entity.
No amount of wealth, which is something Harry regards as normal because it is, to him, makes up for it dawning on him that his brother is the chosen one, and he is not - and that has to be a hard burden to bear.
I am responding to the terms of reference in the OP, which appears to suggest that wealth and so on makes up for the emotional gaps in the lives of the princes, it doesn't.