No, we did not have the experiment. We had two Bush presidents, one lost the re-election and both had to fight in competitive campaigns each time. That meant that they had to sell increased government service, either in the form of welfare or in the form of military expenditures. I simplify, of course, but no matter how popular a politician is, he is still a politician and has to find his client group and feed it, usually by borrowing from someone else’s children.
The point is, we have political dynasties and family nepotism now. Unless you think that this is a great thing in and of itself, I fail to see how making it an official institution could make a bad thing better rather than making it worse.