On the rare occasion when I have to point out
to people that we got rid of our hereditary upper class in 1960, the
perennial response is 'what about the Bush family?'
An hereditary upper class did certainly include the Bushes but has nothing to do with their political success. Name recognition counts and so does political skill which can be passed on to children.
The American hereditary upper class never included people like California Governor Pat Brown and his offspring, N.Y. Governor Mario Cuomo and his offspring, the Daleys in Chicago and certainly not the Kennedy family which made its money in liquor smuggling. Cities and states all over the country have successful family names recurring in electoral politics, almost none from the old hereditary elite.
I think there are two reasons for political inheritance. One is name recognition, but the other, I think is more important. If one elected official is considered honest or at least competent, then the family gets the broad brush of honesty and competence. Offspring get elected from the broad brush.