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.