I tend to think of the "American dream" more like a class lottery. Sort of like the movie Ratatouille, it's not that anyone can be in the upper class, but people from the upper class may have come from anywhere. I think of people like basketball players who come out of poverty. Their mother used to clean toilets 14 hours a day and now that her son is a famous basketball player she can spend her days relaxing in a big house and she never has to clean toilets again.
But the OP is correct. No matter how people made it to the upper class, once they are there their family has an advantage. Even if you start with a totally fair meritocracy where people get their just rewards, because they want to provide for their children (and their children's children) you end up with a class of "old money" who own and control most things of importance.
I'm not sure there is a way around it other than massive inheritance taxes and high inflation, which are historically not super popular ;-)
A different situation is if some wealthy people engineer things so as to effect transfer payments from the middle/poor classes (or just everybody in general) to themselves. I.e., rent-seeking, which obviously exists. While I don't like legacy admissions, afaict it's not rent-seeking.