We've already tried these experiments. Having the surname "Bush" basically guarantees election to high office or at least the GOP nomination. A generation ago, the name "Kennedy" did the same. I'm not convinced the results have been altogether positive.
Nepotism and political dynasties are the WORST aspects of American politics. As bad as they are now, they'd be a thousand times worse if they were institutionalized and made official.
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.