You just made the point here that's the crux of every Rome-China historical comparison. Of course a civilization that controlled as many territories and was as advanced as the Romans were going to leave a legacy. But that's not people are discussing in Rome-China debates because if we were judging the continuity of empires by their legacy then that opens a whole can of worms. As an example, Modern Indians speak English, have a parliament system of government, and at least upper class Indians are educated in the British style, can they claim to be a successor to the British Empire?
The exact point people make is that unlike Rome, China maintained enough of a cohesive internal structure to largely preserve its territorial, linguistic, and core cultural identity for millenia to this day where its one of the world's great powers. What Rome left behind in the Western Empire after its collapsed was the Roman Catholic church, ie. one institution that exercised profound religious authority perhaps but had no head of state, no bureacracy managing territory outside of Rome, and projected little military or economic power compared to even its neighboring city states like Florence nevermind the other European kingdoms. You had barbarian tribes that claimed to be successors to Rome, but whose territories themselves were just a loose patchwork of fiefdoms and duchies that only acted cohesively during times of military campaigns and outside of that often fought eachother. They followed the heed of the Catholic Church, but inherited none of the wisdom or governing efficiency of the Romans needed to administer large territories and promote trade or urbanization. It was wisdom they didn't even have access to until the Arabs reintroduced it to them.
And that's Western Europe, the most ostensible direct offshoot of Rome was Byzantium and we all know how their story ended in 1453.
