One of the big issues with the so-called “Fall of the (Western) Roman Empire” is that it has so very many caveats. Many of the questions remain open precisely because the waters become quite muddy upon inspection - we have to make sure we say that the Eastern Empire kept chugging right along, thanks very much, and that in spite of the ostensible fracturing of the Western Empire, Rome the city hadn’t even been the center of the empire (or the emperors) for centuries… that had been Ravenna… or more accurately, wherever the emperor personally was. Then we get to the issue of all those Carolingian kings and Catholic popes bestowing titles like “Emperor of the Romans” onto guys like Charlemagne… the Holy Roman Empire, the Kaisers, the Czars… when does it really end? All of this makes for great historical debate topics because there’s no real clear-cut, clean answer.
In contrast, the Han had a pretty definite expiration date, or at least a greatly constrained period of “is it gone yet” versus the millennium+ debate over Roman civilization. Likewise its cause of death is still largely agree upon. The Han emperors had been serving as ceremonial figureheads round about the death of Emperor Ling in 189 CE. This had been brought about in large part (as would happen time and again) because of the devolution of imperial powers from the throne and out to the various provincials/regional administrators. To wit: generals and governors had accumulated the preponderance of actual authority, to the point where the imperial structure was used only as a fig leaf. From Dong Zhuo, to Yuan Shao, to Cao Cao, all of the warlords at the end of the Han sought to possess the emperor in order to lend credence to their own directives... culminating ultimately in 220 CE, with Cao Cao’s son, Cao Pi being “offered” the throne by Emperor Xian.
From a formal standpoint, that was the end of the Han. The day before, there had – legally – been a Han Empire – the day after, there was only Wei. Of course, that didn’t last long, since the following year the distant, far-removed cousin and resident “Brave Sir Robin Ran Away” of the story, Liu Bei declared the formation of Shu Han. In this he attempted to use the name of the Han and his own lineage to legitimize his claim. To that end, one might make an argument that – like the Eastern Romans of Constantinople – the Han therefore lived on. But the Eastern Roman Empire had been formally recognized, and had long pre-dated the “Fall of Rome” – all the way back to the Diocletian Reforms. In contrast the formal declaration of Shu Han had virtually none of those trappings – it was self-declared by a, from a legal standpoint, rebel against the legitimate throne of Wei, who had only tenuous ties to the former ruling household. Moreover, in the long run, Shu Han would never come close to doing what a Chinese dynasty must in order to be seen as a legitimate successor state: reunify all (or at least most) of the realm. Shu Han was never able to push out of its Sichuan fortifications, and by the end of the 3 Kingdoms era, it had fallen to the overwhelming might of Wei (immediately prior to Wei itself giving way form within to the Sima Clan and the victorious albeit short-lived Jin Dynasty). The Jin Dynasty would the go on to briefly reunite the whole of China by conquering Wu to the southwest, thereby sealing its “victory” over the Three Kingdoms (for whatever that was worth… before the realm was plunged back into another 3 centuries of chaos and strife in the form of the 16 Kingdoms and the Northern and Southern Period.
So, in essence, when the Han died, it stayed dead… and there really wasn’t anyone holding onto the thread of its recovery or continued existence post-220. Sure, Liu Bei sought to capitalize on his link to the Imperial Liu Clan to bolster his own power… but even that wasn’t seen as an attempt to “restore” the Han… but rather to replace it with the Shu. And in the end, regardless of how one might interpret Liu Bei’s motiviations, he failed in any case. Thus, at some point in the 3rd century, whether we want to say 220 or 263, the Han Dynasty was eradicated totally and permanently… and then it was buried under 3 centuries of civil war and would-be kingdoms rising and falling.