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The short answer for Democrats is easy: the end justifies the means.
Look back at the past 35 years or so. From Roe v Wade forward, the left has been using the courts--the least democratic of the 3 branches of government, to accomplish its ends. And the reason they pursued this course is because they knew that the democratic approach--where the majority would make the decision, would not work. Quite simply, when it comes to a vote where the majority wins, liberal notions generally do not do well.
And once you get on that anti-democratic road, you are guided by the end-justifies-the-means principle in other contexts as well. And since powerful political families are easier to elect in the sound-bite world we live in (name recognition) and because they can achieve the end being sought as well as or better than unknowns, they get support from the political powers that be.
Of course, there are other reasons that political families do well, including some positive ones. The Kennedys and Clintons are a good example--when smart people marry other smart people and then raise a family of other smart people--with a bit of initiative all of those smart people may pursue politics in order to accomplish their goals. That can be a good thing for the country.
By the way, in states that have term limits, these political family dynasties are widespread. I have seen one legislative seat in my state go from husband to wife to son and seen others move from spouse to spouse when the first spouse moved to the Senate from the House.