> the Holy Roman Empire inherited the name

Eh, inheritance implies the predecessor willed it to you upon death. In the case of the "Holy Roman Empire" the predecessor died 334 years earlier with no successor. This isn't a case of inheritance, it's a case of just making shit up.

The "byzantine" empire did inherit the name though. They continuously used and claimed the name "roman" from when the roman empire split (and called both halves roman) right up until 1453, even their language was called romaic. That we don't refer to them as roman to this day is really a strange success of propaganda.

> And the Byzantine Empire shrunk greatly after the first few hundred years.

Which by itself sort of contradicts the claim that empires historically fall quickly? The roman empire began in 27 and in 565 it controlled almost the entire Mediterranean. Sure a few hundred years after the split (which itself is a few hundred years after the beginning of being an empire) it shrank but that's a long time.