the question of "when did Rome fall" is, of course, flawed in its initial assumptions. Rome didn't fall in 300 and it didn't fall in 476 and it didn't fall in 1204 and I think you could say it fell in 1453 unless you say that the Ottomans were just incorporating Constantinople, in which case I'd say it didn't fall until the 1900s. But that's all a pointless exercise unless you like to rattle off dates, which is a fun sport for all ages.
But in my opinion the best way to judge when an Empire falls is when people switch their viewpoint from wanting to make the Empire theirs (or to make the Empire better, or to make the Empire closer to its "true form") and they start thinking in ways of making their own Empires that are actually a different entity than the Empire.
Thus, I always think that the real "moment that the [Western] Roman Empire fell" was when the people on the borderlands started saying "actually we don't want to take over the Roman Empire, we just want to be the Goths, who are not the Romans". And yes, of course, they integrated elements of Roman-ness into their Empires, because that was just kind of a smart thing to do (the Romans certainly did plenty of it), but it's a different mindset, you know.
The Empire lasts until it dies in the minds of the people who are harmed by the Empire. The Empire lasts until it is supplanted by another entity that recognizes the Empire as an enemy, not as a crown to steal. IMO, that's the moment that "Rome" "fell".
