"The parents kept the money they earned, and (quite rationally) chose to spend it on giving the children an economic advantage (that the children, more often than not, could not have earned for themselves.)"

So the children, who could not have earned that money themselves, should get it anyway? Why are they particularly special -- why not redistribute money to other people who have trouble earning their way? Why does the fact that you are someone's descendant someone exempt you from having to earn your way just like everyone else?

That depends whether you agree with the parents having the choice of how the money they earned is used. My point is that if we take the money from the children on the grounds that they do not deserve it, we implicitly deny the parents that choice. Which may be what you want. But if you do give them the choice of how to spend what they earn, you must accept that they may choose to spend it on people (or charities, or products etc.) that you consider undeserving.