600 BC is pretty far back. They thought they were descendants of Trojans who escaped the destruction of Troy.
What would have been more interesting would have been if Romulus was in it.
That was the invention of Virgil to tie Rome to the great epics of Homer and give it a national legend all its own. And to curry favor with Julius Caesar and/or Augustus by renaming Aeneas son Julius, implying the family Julii were the founders of Rome.
As I understand it, and I'm just an armchair reader of history so take that caveat to heart, Romans of the time more generally believed that Romulus founded Rome and it fell under Etruscan rule in the mystic past until there was an uprising that freed the city from Etruscan rule led by one Brutus, and this played into the pressure put on Brutus' descendant centuries later to help kill his own mentor Julius Caesar. The symbolism of a Brutus once again liberating Rome from the grip of a tyrant made for good optics.
There is archaeological evidence of human occupation of the Rome area from at least 5,000 years, but the dense layer of much younger debris obscures Palaeolithic and Neolithic sites.[2] The evidence suggesting the city’s ancient foundation is also obscured by the legend of Rome’s beginning involving Romulus and Remus.
The traditional date for the founding of Rome is 21 April 753 BC, following Marcus Terentius Varro,[3] and the city and surrounding region of Latium has continued to be inhabited with little interruption since around that time. Excavations made in 2014 have revealed a wall built long before the city’s official founding year. Archaeologists uncovered a stone wall and pieces of pottery dating to the 9th century BC and the beginning of the 8th century BC, and there is evidence of people arriving on the Palatine hill as early as the 10th century BC.[4][5]
After 650 BC, the Etruscans became dominant in Italy and expanded into north-central Italy. Roman tradition claimed that Rome had been under the control of seven kings from 753 to 509 BC beginning with the mythical Romulus who was said to have founded the city of Rome along with his brother Remus.