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A single article in Time Magazine from 1929 isn't persuasive enough to draw the conclusion that the problem of nepotism is worse today than it was in the past. A telling quote from that very article suggests quite the opposite: "[m]any another son has followed his father into high office."
Somebody's going to have to do or find the research comparing the degree of nepotism in today's political class compared to that of yesteryear's. Only then will we know in which direction we've moved or if we've moved at all.
As for "what accounts for this anti-democratic dynamic," I'd posit human nature. In the Roman Republic, it was quite rare for a novus homo ("new man", i.e. one w/out a family history in politics) to get very far. Things have certainly improved since then, but I'm not convinced that they've worsened since the birth of our own republic.