And yet, there are 800 million native speakers of Romance languages ("the modern languages that evolved from Vulgar Latin between the third and eighth centuries") in the world [0].
Instead they speak direct descendants of the language of the common tongue of Roman citizens, 'vulgar' Latin. The pre-Roman languages of most of Roman occupied Europe were extinguished. Apart from Greek, which was kind of the second language of Rome (and later the only language, once the west collapsed) and Palestine/Syria, it's only in Britain where pre-Roman languages survived -- that is until the Anglo-Saxons wiped most of it out, excepting Wales and Cornwall.
It very much did.
> never mind Europe.
It did that, too, though less tightly.
> The lands and tribes they conquered simply used Latin for trade and as a lingua franca. It never became the mother tongue for any of those people.
And where exactly did the Romance languages that are the mother tongues of much of the area that Rome used to rule come from, then?
Italy, France, Spain, Romania, Portugal all speak a variation of 'Latin'.
It'd be more like saying Finnish or Hungarian are a variation of Latin or German. Some small amount of shared vocabulary (and maybe phonetics) due to proximity, but no more.
This is simply false. Please learn about the pre-Roman history of Italy.
Not really. The Chinese tend to emphasize viewing their history through a lens of continuity and a single notional empire whereas the Western history tends to view their history as discontinuous and fractured empires. But when you start comparing the actual effects of these empires and their rises and falls, they are actually much more similar.
For example, after the Crisis of the Third Century, the Roman Empire was effectively partitioned into a Western and Eastern Empire. The Western Empire proceeded to decentralize itself out of existence over the course of the fourth and fifth centuries, but it recoalesced under the Merovingian and Carolingian dynasties into the Holy Roman Empire by 800 (before splitting up into effectively a Frankish kingdom and the Germanic kingdom of the HRE). The Eastern Roman Empire lasted until around 1200, when the Fourth Crusade conquered it instead (a conquest which didn't last, but the Byzantines never recovered and the Ottomans finally put them out of their misery 250 years later). It's worth noting that pretty much all of the people I've cited considered themselves the true heirs of the Roman Empire.
Meanwhile in China, the Han dynasty crashed hard in the early 200s, with three major successor states cropping up. One of these managed to conquer the others, so the successive dynasty was the Jin dynasty. Of course, the Jin last control of most of their territory, which fragmented into the Sixteen Kingdoms, but the entire era is still called the Jin dynasty. After the final collapse of the Jin, the north/south dichotomy continued with the Northern and Southern dynasties, which is again a successive series of powers vying for control eventually coalescing into the Sui dynasty after one wins out, which quickly transitions to the Tang dynasty, the first truly unified China since the fall of the Han dynasty. After about 300 years, the Tang dynasty collapses into a major period of disunity, the next truly unified China again taking 300 years to crop up again.
There's a lot of similarity. The major difference is that Western Europe never reunified into a single unitary state (unless you want to count the extremely short lived Napoleonic Empire and Third Reich). But Western Europe did have a successor state that directly retained the traditions from the Roman Empire (namely, the Byzantines) and could plausibly claim to have no break in continuity (the Byzantines never called themselves anything other than the Roman Empire), and all of the major successor states directly drew inspiration and customs from the Roman Empire, rather like China. The writing system, the state religion, the bureaucratic apparatus, the legal system--all of these are derived from the Roman Empire in Western Europe, just as much as they are derived from the Qin and Han dynasties in China.