Despite the spin from post-enlightement historians recasting everything between ~600-1600 into the 'dark ages', 'rome' as in 'the roman empire' didn't end in the east until 1453 (or possibly 1917, depending on how Romanov you go), and didn't end in the west until 1806 (or possibly 1914, depending on how Hapsburg you go).
Not 'ancient rome', but still self-styled as 'rome' with a direct lineage in some form or fashion.
Polemic aside, certainly cappadocian caves would fall under the 'eastern roman empire' category, albeit post-pagan as you point out.
As Voltaire quipped: "The Holy Roman Empire was neither Holy nor Roman, nor an Empire"
Just assuming the title of the Roman emperor is not really a direct lineage.
But in those 324 years a lot happened that make the case of a direct lineage questionable. The Gothic wars destroyed a lot of remaining infrastructure and Roman culture, the Roman senate was no more after 600 CE, the long term population loss, and the Christian Pope was the defacto ruler of the city (if you could still call that what was left a city).
The coronation of Charlemagne was also considered illegitimate by the Eastern Roman emperor at first, until they couldn't ignore Charlemagne's military power anymore.
It's not any more or any less the end of the Western Roman Empire than the political troubles that occurred in the earlier and later centuries. The Gothic, Frankish or Vandalic kingdoms very much thought of themselves as belonging to a Roman world.