You'll note a declining example of usefulness in your examples. The Byzantines were Romans, I would agree to that example without reservation. However, there's a strong point of difference between the Byzantines and the Chinese. The Eastern Roman Empire was a latin overlay on Greek structures, and even though the Western Empire had its heart captured by the Greeks, the Eastern Empire really was Greek and instead had its heart be Roman. Though there is a clear, unbroken line of Roman institution until 1204, by that point they could rightly have claimed to have given up on most of what it meant to be Roman by that time. The others are, frankly, wrong. Charlemagne comes close-ish, but its more of an alternate history scenario to use him rather than a real example.
China's Byzantine equivalent was the Eastern Jin. Jin was formed by a palace coup when Sima Yan deposed Cao Huan, after he and his family had risen and killed off a number of Wei Emperors. Cao Huan was the last imperial descendant of Cao Pi who was the son of Cao Cao, the Prime Minister of the State and the Prince of Wei, and founder of the dynasty when he deposed Liu Xie who was his brother-in-law. Liu Xie, likewise, was the last imperial son of the Han Dynasty after it was restored by the Guangwu Emperor Liu Xiu. The Guangwu Emperor restored the dynasty after a non-Liu relative the the throne, Wang Xian, formed the short lived Xin dynasty after deposing Guangwu's cousin. Eastern Jin continued with this tradition. The argument that the Eastern Jin is only holding the myth of being China is frankly bizarre. The Southern Dynasties were brought to an end by the Sui, a man named Yang Jian. He rose to power in a way that mirrored the above, and took power when his son in law, the last emperor of Northern Zhou, died. Unlike his son in law, who was a sinicized descendant of northern martial migrants who conquered part of that land, Yang Jian was actually what we would call Han Chinese. However, the Northern Dynasties had become Chinese in the Byzantine sense of being Roman a long, long, time ago. Only instead of using Greek, they used the Imperial Script. Rather than becoming more and more into their own, like the Byzantines did, they became more and more like what was there before.
Yang Jian's Sui was shortlived, but a relative of his founded the Tang under the Li family. This was the second Golden Age of Civilization. Later transition zones were less complicated and less destructive, and overall never really threatened the existence of China. The Song emerged rather rapidly from the ashes of the Tang, as the Five Dynasties and Ten Kingdoms Period was something of a free for all, and saw the emergence of some people trying to restore earlier dynasties that had ended with their ancestors. The Song Dynasty had only one major rival at a time (Xi Xia was small and weak, Western Liao could only generously be called Chinese-ish), the Liao Dynasty and later the Jinn Dynasty. These were Northern Dynasties, though a little more tangential to China and thus a little less drawn into its orbit. At least, until the Jinn captured Kaifang. The Song only recognized the Jinn when defeated in war, and often sought opportunity to expand North and reclaim the northern district which they saw as Chinese, and were Chinese. The Song allied with the Mongols to eliminate the Jinn, but the Song misjudged the Mongols by thinking they would just be another Northern Dynasty that could be dealt with, and maybe even pushed out of China entirely.
The Yuan Dynasty presents the first actual break in Chinese history, and they were overthrown by the Ming. However, the Yuan kept most of the Chinese government in place, along with much of the scholar gentry. This is why the actual Chinese powers sided with the Yuan Dynasty against the rebellions by lower classes, and only made the switch when they were powerful enough. Once the gentry turned on the Yuan, they were pushed entirely out of China. The Northern Yuan remained a substantial threat, but the Ming continually played on tribal politics and it was kept relatively weak. It was finally destroyed by the Manchurians, who were in the process of becoming the Qing Dynasty and had originally claimed the be the Jinn Dyansty restored. Manchurians either are or were related to the Jurchens who had founded the Jinn. The Qing Dynasty was invited into China by Wu Sangui following another dynastic transition period where peasant rebellions and new dynasties were coming to power and in conflict with the Ming state. The original intention behind the new Jinn State is unknown, but as high ranking and influential Chinese generals and civil officials defected en masse to avoid Li Zicheng and his Shun Dynasty. Famine was killing millions, and many were desperate for steady leadership and peace. The Manchu's discarded the Jinn name a few generations in to exchange it for the Qing, and proceeded to conquer the entirety of China rather than the northernmost part. And the Qing kept Chinese in power, and were far more Chinese than the Yuan. Eventually the Qing feared that the Manchurians would die out completely, and took measures to keep the Manchurians separate. Less than a generation after the Qing Dynasty fell, Manchurian language nearly disappeared and is now a dead language and the Manchurians have largely become indistinguishable from Han Chinese.
Now, I don't meant to say that the Chinese did not engage in historical propaganda. They did. But you may have it backwards. The Chinese were not fabricating links to the past, they were trying to make the connections more overt and justify the formation of a new dynasty or conquest. Famines and rebellions were in almost every dynastic transition by conquest, while incompetent, weak, or evil rulers tended to precede every dynastic transition by coup. That this was so common and appeared so often in the historical record, it is believed that the historians from the scholar gentry were basing their histories on what they believed had to happen for Heaven's Mandate to be passed on to another family, and exaggerated other events. At the same time, that there were bloody transition zones is readily apparent, and the Age of Fragmentation remains a conflict that was more bloody than almost any other conflict in world history and did very nearly break China if not for my first paragraph.
Not to mention that states that failed to actually recapture the entirety of China went into propaganda overdrive in order to justify their existence when it was clear that heaven's mandate was neither one nor the other. This is particularly true amongst states like Eastern Wu or Shu Han. Its also no doubt that this national myth was a key part of their identity. Chinese call themselves Hanren, because they see themselves as the people of the Han Dynasty. China may not have been a term yet, but People of the Han Dynasty was in use for almost two thousand years. Hanren also call themselves Tangren, because they are people of the Tang Dynasty. Minority groups, particularly in the south, have a preference for Tangren over Hanren, particularly when Hanren became the name for the main ethnic group of China.
I recommend learning about the foundations for China's Byzantine Empire by reading Professor Rafe de Crespigny's
Generals of the South, and then explore more about China and its history.