I was reading the other day how most of the new testament was written in Greek rather than Aramaic due to the fact that at the time Greek had become the universal language the different people from the eastern part of the roman empire used to communicate amongst themselves. That left me thinking, why didn't the Romans try to force Latin to be the main lingua franca rather than Greek? Wouldn't this have made it easier for administration purposes and created a stronger sense of unity with the rest of the empire? I imagine that greek had already taken this role before the Romans had arrived but couldn't they have tried to change things up?
Greek literature is hundreds of years older than surviving Roman literature. Homer's epics were written down sometime around 700 (it's still a hot debate); Greeks had a colonizing habit, making settlements in Anatolia, Sicily, France, and Magna Graecia (Italy), some even in the Mycenaean period. Cyrene was founded in 631 in Africa. Massilia about the same time. Syracuse in the 700s. Thapsos seems to have been founded in the Mycenean period.
And then you had Alexander's campaign through Anatolia, the Levant, Egypt, and Persia. Along the way he founded some 70 cities during his decade-ish long campaign on the Greek model, to settle at least some of his Macedonian soldiers, effectively bringing Greek across the near and middle East. That was in the 4th century.
By contrast, the first known Latin prose writer was Cato the Elder, who both knew Greek and actively opposed its use in Rome (so Greek had already penetrated Rome). He died in 149, quite a bit after Alexander. 200 years almost. Prior to Cato, Roman history had been written in Greek. Quintus Fabius Pictor. There's something to be said for the notion that "history" as we think of it was a Greek genre (Herodotus, Thucydides, Xenophon), and Pictor chose that language to fit in with the genre. Cato very much disliked his son learning Roman history from a Greek text, according to Plutarch, and so wrote a prose Latin history of Rome for what seems to be the first time.
There is some earlier Latin poetry. Something called "Saturnian" which we don't understand very well. It seems to be alliterative in some way. Others stress assonance. But it remained local, and died out as hexameter became more popular in Rome. I'm familiar with the Saturnian stuff mostly from funerary inscriptions. But we're still talking 4th century for the early examples, I think.
Where Latin did take hold were places the Greeks had spent less time in. France, Spain, northern Africa, northern Europe. But that was centuries later.
The Greeks just got everywhere, including Rome, first.