I like the term "Byzantine Empire" specifically because it sounds cool. However, I maintain there is not some point in time where you can logically "de-Romify" the Eastern Romans based on some event/trend/factor. As I have posted before on this topic, being "Roman" by the Fifth Century AD was no longer AT ALL descriptive of Italic ethnicity, speaking Latin or being from the city of Rome (or Italy). It just wasn't. People had begun to view Rome as the "one true empire of God on Earth," and being Roman was entirely dependent on fulfilling that vision and carrying on that legacy. If the US split into the Western US and the Eastern US, and the Eastern US (i.e., where the original English Americans started out "civilization") fell, NOBODY would consider the Western Americans as not just "Americans," even if they ended up speaking Spanish in this future scenario.
The difference is that the US isn't called the United States of Washington DC (it has the opposite problem of having a too generic name rather than a too specific one
). It's really just a semantic issue, as I already said that the Byzantines are absolutely a political and cultural continuation of the Romans. But there is something weird about calling an empire over a city that it only held nominal control (and later no control) over and which had no real political relevance within it. And I'm aware that the political decline of Rome within the Empire started long before the fall of the West. That's why if anything I'd be in favor of retiring the term "Roman Empire" earlier. But again, it is just semantics.