It's been awhile since I was reading into this stuff, but I believe the idea is that these familial success correlations last a
long time (Maybe multiple centuries? Certainly at least one) where as money and other forms of success (which are the things we are actually measuring) have significant noise on much shorter timescales. Those fluctuations would kill the long-term correlations if money were doing the heavy lifting.
In other words, if this were being transmitted through money, you'd just condition on the parents (or grandparents) amount of money and expect to find that the success of children to then be independent (unless, e.g., children were all digging up gold secretly left to them by their long-dead great-great-grandparents). But instead, I think the idea is that even after you condition on success of parents and grand parents, the success of great-great-grandparents still has strong predictive ability.
Indeed, if you thought money was playing a significant role, you'd expect that exogenous injections of money would lead to long lasting effects, but in fact they decay very quickly.
http://slatestarcodex.com/2014/05/26/compound-interest-is-th...
That still allows for the possibility that money is buying something that one's ancestors can robustly transmit across many generations (often bundled up under the term "culture"), but then you could test that hypothesis with twin studies and adoption studies.
Anyways, that's a cartoon version, and undoubtedly simplified and biased by the small number of people I was reading. Would love to get counter data.