Yeah, but there are lots of modern groups which claim to be, and maybe are, continuations of old ones, but people don't consider them to be. Is the United Kingdom still the British Empire of the 1700s? The Byzantine Empire was at most a rump state clinging to something it lost. I mean, how can it call itself Rome if it doesn't even hold the city, for the most part/
So, I guess that's my real argument; how can there be a 'Roman Empire' that doesn't hold Rome?
That's a good argument.
Well whenever you're discussing what was or wasn't Roman, you have to keep in mind the general arc of roman history up to 476.
1. It was the city state of Rome and its clients.
2. It became a pan-italian state ruled from Rome.
3. It expanded into a Mediterranean wide empire.
4. Other bases of power than Italy took on increasing importance.
5. Power shifted east. State and economic structures in the west began to dissolve.
6. The empire was reduced to its eastern, Greek possessions.
The transformation of the city of Rome from the center of the Roman world to one major city to a backwater is a consistent trend in the evolution of the Roman state. That's why I don't think it makes any sense to mark its loss as the end of the empire.