It's had to imagine the USA ever legalizing the idea of a "royal family" or any type of hereditary nobility.
In the south and middle part of the country, it would be hard not to imagine seeing the monarch of the united state as an agrarian aristocrat with a premium placed on blood and wealth. The north prefers the orator, merchant, and lawyer - but even they had inclinations to be snobby when it game to heritage. All the way up until near modern times, the citizens of the United States cared a great deal about your parents, grandparents, and so forth - how and where you were raise, military service, how well you can speak, and how much money you have. Hell, those are still highly recommended marks of being a public servant.
That, and most British people thought the American Revolution would eventually settle into a system much like their own in Great Britain. It took a few decades for those ideas to die out as the older generation was phased out and the people began to trust in the republican government. Presidents stepping down after two terms and politics being a peaceful transfer was, perhaps, the most pivotal mark of the new democracy. Just by having it be for life may inevitably lead to a pull in the opposite direction as people look more towards Europe, especially with European immigrants flowing into the country.
It is thought that the young nation believed they were intellectually superior, but culturally inferior to Europe until the twentieth century. Should this new government keep the intellectual rhetoric, but more closely align its culture to Europe, then the changed nature of the "presidency" or elective monarchy can easily culminate in one family taking on de facto, if not de jure, control over the nation.
After all, the French Revolution was in many ways more radical than the American Revolution with people who had even greater reasons to question the regime than the taxes that were almost mere pittances on the people, and an occupation that was tame by almost any contemporary standard. They fell to monarchy three times, five republics, and a handful of intermediary dictatorships. I honestly think people do not realize just how significant the United States's history really is as a peculiar event in human history.