This may have been true in the past, when people tended to have lots of children. But with families having fewer children, it is probably less likely to be true in the future. Just as a personal example, I have 2 siblings and over 10 cousins. My daughter has 0 siblings and 1 cousin. Those numbers are going to go up by one or two but not much more than that as the window for kids is quickly closing. My family is solidly middle class, but it is entirely possible that that my daughters generation of the family ends up as "trust fund kids" simply due to the lack of dilution between siblings and cousins. Also, it helps that estate planning is significantly more accessible than it used to be.
Regardless, the OP was not necessarily talking about the generational wealthy, they quite literally said the "idle rich". And there will always be a fair number of those folks, regardless of whether it was multi-generational or not.