I wrote an answer on how the Romans reckoned dates here. Much more than we do, they worked with relative dates - so things like 'this happened two years before Augustus became emperor', or 'ten years after the fall of Carthage', using well-known reference points. Both the BC/AD system and the AUC system you've mentioned are basically extensions of that impulse.

AUC wasn't used very much in Roman times - you do see dates reckoned in relation to the founding of Rome, and they did have a more-or-less consensus idea of when it happened, but by far the most common way way to use the dates of consuls. There aren't very many cases of Roman writers needing to talk about things happening much before Rome - I'll chance my arm and say that every surviving Roman history book (excluding those written by Greeks in the Roman empire, which I know is a bit of a cheat) deals with specifically Roman history.

There's also the issue that most events before the founding of Rome are back in quite vague mythological time, something of which Classical-period authors were well aware. Here's an illustrative quote from Plutarch, at the start of his biography of Theseus and Romulus:

Just as geographers crowd the outer edges of their maps with explanatory notes that 'what lies beyond is sandy desert without water and full of wild beasts,' or "blind marsh," or "Scythian cold," or "frozen sea," so I might well say of the earliest periods: 'what lies beyond is full of marvels and unreality, a land of poets and fabulists, of doubt and obscurity.'

You could certainly work backwards from known dates - but often the Romans simply didn't bother. When Livy writes the earliest, pre-Romulus history of the Romans' ancestors, for instance, he doesn't include any dates - he says that Aeneas, their legendary ancestor, fled from Troy, and gives the sequence of descendants and rulers that take you from him to Romulus, but he and his readers were much more interested in the names and deeds of the leaders involved than trying to pin exact dates on everything. So too with Tacitus - except he talks about the order in which systems of governments evolved and were used, because he's interested in expressing how imperial government represented a coming-back-around of the earliest monarchical system.

It's important not to get things backwards - the fact that these writers avoided precise dates wasn't a matter of them being forced to do so because of a clunky dating system. Instead. the dating system the Romans used, with its advantages and disadvantages, reflects the priorities that they had, and exact dates in deep time were not really among them.