This question depends as much on you as it does on history, because it's about identity: what made Rome Rome? To say "Rome fell and nobody noticed", you have to have a different definition of Rome - or maybe we should say, assert a different essence of Rome - than the people who lived at the time.
I'm mostly arguing that people overindex on the political changes and don't see the ground level continuity. Map color changed != they're not Roman any more.
Regardless, I do enjoy asking "what made Rome Rome?" — I like the idea of the Byzantines being Roman in 1453 even though at that point it was tenuous.