That's a key difference, however.  America's founders  mythologized the Roman republic.  European rulers  mythologized the later Roman dictatorship. ("Czar" and "Kaiser," for example, are local pronunciations of the informal Roman military title in the last empire, "Caesar.") During all of Rome's period, even the later dictators and their historians mythologized the Roman Republic and Athenian democracy and never employed a formal title like "emperor." (There was never a formal institution for the ruler of the empire in Rome until well into the Byzantine period -- just an informal habit started by Augustus of selecting a leading general of the armies who, as a private citizen of civic will was expected to protect the Roman republic. Kind of like if Warren Buffet were suddenly granted command of a huge private army to replace the poorly managed Pentagon and in return was expected to ensure that politicians uphold the principles of the US constitution.)

Rome was a much less formal affair, rooted in the contradictions of have been a republic, than monarchies of Europe that sprang up afterward claiming divine rights and the titles of kings and emperors to rule and such.  My argument is that America is much more like Rome in that sense as well, as any of the European empires with which it has been compared.