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There is so much of interest in this article and the comments that it is impossible to address all of it.
A couple more families in politics: The Lodges from Massachusetts, and the Stevensons from Illinois. Also, I don’t think it was mentioned that two of FDR’s sons served in the House of Representatives.
There is probably a master’s thesis that could be earned by examining this trend in politics, and determining whether it more common now than in the past.
I don’t think it is uncommon for sons or daughters to follow in their parents footsteps when the vocation is rewarding, whether financially or emotionally. You see many ministers who are sons or daughters of ministers, and of course it is a cliche that many businessmen want their children to go into the family business (you could also include organized crime in that category).
I think the reason that nepotism is so common in the entertainment field is primarily opportunity. I am not saying that acting is easy, but you can go to virtually any regional theater and see actors who are at least as good as many big stars. Children of stars have the advantage of familiarity with the business and if they have any talent at all will be given at least a chance.
I agree with others that political nepotism is about branding. If you vote for a Kennedy, you have at least some idea of what you are getting. That doesn’t really explain why people would vote for George W. Bush after rejecting George H. W. Bush only 8 years before, except that people often pick the familiar over the new, even when the new is demonstrably better. A good example of this is major league baseball, where managers will often play a mediocre veteran over taking a chance with a promising newcomer.