Agree with sentiment but want to nitpick a bit. It’s unclear that the Roman Empire collapsed at all. Many Roman institutions lived on well past the 5th century and were assimilated into gothic Frankish Visigoth etc societies.
https://davidgalbraith.org/trivia/graph-of-the-population-of...
The etymology of Zhongguo IIRC is "central city". Was this the actual name of an actual city that then became the name for the concept of a central city/state as dynasties and rules changed? Is it analogous as if the word Roma had become a generic term for state in european languages and now you'd just call the United Romas of America?
中国 (Zhōngguó) is middle country/state/nation, not city.
I don't think Chinese is that flexible. The only thing I can think of is Chengdu (which is not really related): https://en.m.wikipedia.org/wiki/Chengdu#Name