Well they did have more normal heredity within dynasties, the emperor would have to lose the mandate of heaven before someone else could get it, it's not like you could just kill the emperor and sit on his throne and expect to take his position without a great big war, and even then you'd expect to lose control over outlying areas.
And China very much did splinter and collapse in a matter of generations as well. The idea that there's only ever been one China is essentially propaganda by recent centuries of Chinese nations who can no longer rule by right of being the emperors due to not being that kind of government.
Assassinating him and sitting on his throne, no. But if you take over China via war,
by definition the emperor has lost the mandate of heaven. Because how could anyone with the support of the gods lose? In the case of natural disasters or a terrible ruler, some people might think that means he's lost the mandate and take it as an opportunity to rebel. You don't actually know until you see if the rebels win tho. Because if the emperor manages to defeat a rebellion, by definition the gods supported him over the rebels. Whoever holds the throne deserves the throne, that's what the mandate means. Its part of the Confucian ethic of loyalty and obedience. Under Confucianism, the only time you're supposed to rise up against your rulers is if they're terrible, not merely bad. The mandate justifies this doctrine by providing a blanket explanation for why any ruler should be obeyed.
As for China splintering, you're not wrong. It gets down to definition of what "China" is, I guess. Was China before the Three Kingdoms Period and after it, for example, the same continuous nation state that temporarily underwent civil war, or two separate things sharing a name? I would count it as continuous but that's debatable.
@Descan: I dont think that's true. Mongols assimilated to Chinese culture because most nomad invaders largely assimilate to the places they settle down. The same thing happened with the goths.. and the Normans too, who were French-assimilated peoples in the firsr place and in turn assimilated to English, Irish, and Scottish peoples
IMO you have that backwards. The Goths (and other "barbarian" migrants/invaders) effectively supplanted Roman culture throughout most of Europe and then turned around and pretended their civilization was descended from the Roman one. The Normans effectively destroyed Anglo-Saxon culture after their invasion. Then they tried to erase Irish culture in the same way, but failed because their first wave of lords assimilated and learned Gaellic. Then they tried to cut out the middleman and just wipe out the Irish.
I would also point out that the Mongols adopted local culture pretty much nowhere except China. The closest they came was to adopt a hands off approach to governing some of their territories, but in those cases it was locals who were doing most of the ruling.