First, Rome really didn’t fall, as such; or at least not the way generally thought, or when generally thought. Certainly there were times when the barbarians were literally at the gates, so to speak. But there was never a time when a Roman citizen of southern Gaul, for example, would stop and say, “My, haven’t things been crappy since Rome fell.”
But by the time Rome “fell,” it had already been superseded (and eclipsed) by the New Rome (Constantinople). These Romans of the East thought of themselves as exactly that, the “Romaioi.” They would have been puzzled if anyone had addressed them as a Byzantine. And not only that, but this New Rome — for centuries, the marvel of civilization-continued on until 1453. So, even if one were to buy into the whole America = Rome theory, the date used for comparison is about 1,000 years shy of the final fall.