Just my opinion but a lot of this has to do with parenting skills and instilling the right values into your kids. It doesn't matter how much money you have, your kids could grow up poor and greedy, or rich and greedy. They can grow up wealthy with a work ethic and drive, or they can grow up poor and lazy. In the end it's not about the money, but the values. These can be taught and while every child eventually has to choose their own path, it is typically the one that they were brought up on.
So what I'm saying is that we should teach our kids to live simply, work diligently, and give generously. Whatever money is passed on to them should not change who they are and what they will do with it. If we've trained them well in what money is, and what it's supposed to be used for, then a lot of the problems that most of the snobby rich kids have should not show up in our families.
Personally, I want to be a billionaire. Set up a private foundation so that most all of it is given away, and if my kids want to run it for a modest salary after I'm gone, then my wealth will have provided for their basic needs but not all their wants. There's a saying at my church that we want to provide for your needs, but you need to trust God for your wants (read: work for it kid).