A lump sum of money and an estate built around extracting wealth from economic processes are two different things. The typical lottery winner doesn't have the social connections or technical skills necessary to create such an estate themself. Having money quite obviously is different from having investments.
If the lottery winning family does not focus on wealth accumulation, then their wealth will regress towards the mean for their lifestyle. If the lottery winner puts all their winnings in an index fund, that would be a good start to creating such an intergenerational estate.
However, surprising perhaps to many people, not everyone is interested in turning money into more money. People who try to make money, surprise, have the skills, connections, and desire to do so. Of course, wanting to be such a "self-earner" doesn't mean one has the means or the luck to become one.
The key point is this: if your estate is currently extracting wealth from investments, odds are that you inherited that estate. Your family having money in the past is a precondition of such a state, but it is not sufficient to sustain it. Wanting to be the first generation of your family with wealth extraction privileges does not mean you'll succeed and most such aspirants fail.