Ancient Rome is often seen as a collection of small farming settlements dominated by an Etruscan elite. We do have archaeological evidence for the existence of kings at Rome, but we simply can’t tie that evidence too tightly to the traditional narratives. There is a long gap (roughly three hundred years) between the end of the monarchy and the beginnings of Roman historiography.
Early Rome had a significant Etruscan population, and it was not unusual in that part of Italy in general for great families to move around and assume positions of power elsewhere. There is evidence of important Latins in Etruscan cities, and of course much evidence of Etruscans in Rome.
The archaic Rome was in the border of the Etruscan and Latin territories, and it seems indeed to have benefited from some mixed identity and a lively interchange between both cultures. The demography of ancient Rome was thus multi-ethnic, including, for example, Etruscans and other Italics besides the Latini, the latter seemed to be the dominant constituent. Actually, the Etruscan exerted a strong cultural and political influence on Latium populations and differences between Latini, Italics or even Etruscans were probably less important than we thought.
The story of Tarquin the Elder, descendant of a Greek refugee who came to Rome from an Etruscan city, well reflects this multicultural situation and the Greek influence. When the younger Tarquin had need of definitive oracular advice, he sent his sons to Delphi in Greece, and it was to Greece too that the founding fathers of the Republc went for the laws that were to be embodied in the Twelve Tables.
The prevailing view is that Rome was founded by Latins who later merged with Etruscans. In this interpretation, Etruscan cultural objects are considered influences rather than part of a heritage. The main criterion for deciding whether an object originated at Rome and traveled by influence to the Etruscans, or descended to the Romans from the Etruscans, is date. A second criterion is the opinion of the ancient sources. A modern theory of etymology holds that the name of the city could be of Etruscan origin (and perhaps the city itself, though this cannot be proven), derived from rumon, "river".
Traditionally we do consider that the Romans absorbed much of their culture from the Etruscans, whose culture was influenced by Greek resident traders and by the Greek colonists of southern Italy. But when you have a map of Greek colonies and ancient trade routes, you can see that Italy and Latium are situated in the middle of complicated network of trade routes and colonies. Koine Greek had become one of the most common trade languages of the Mediterranean and into Asia Minor after the conquests of Alexander the Great.
There was much contact between Etruscans, Romans, and Greeks in Italy when Rome was still a smallish city-state. Rome was also the intersection of the principal roads to the sea coming from Sabinum (in the northeast) and Etruria (to the northwest). Greek traders resided in Rome, alongside many traders of various origins. The Greek influenced all of Italy and probably directly influenced Rome too. Major Greek towns like Naples and Cumae are about 200 km south of Rome, or 10 hours (by trireme) to 25 hours (merchant ship) away.
We can still consider that before the Pyrrhic War (280–275), the Greek-Roman relations were mostly one-sided as Romans were seen as Barbarians. The Greek accounts about the Romans changed after this war. The Pyrrhic War was Rome's first confrontation with the professional armies and mercenaries of the Hellenistic kingdoms in the eastern Mediterranean. After this victory, Rome was recognized as a part of the Mediterranean diplomatic system and asserted her hegemony on Magna Graecia (southern Italy).
It was at the time of the Macedonian wars of the early second century BC (214–148 BC) that the Latin cultural tradition was exposed fully to a culture that the Roman elites considered in many ways as richer and more sophisticated. Rome became heavily involved in Greek affairs and the Roman aristocracy seized upon Greek culture.
A nice POD, would be Pyrrhus winning against Rome and traumatizing them as much as Brennus did. It could lead to a conservative and reactionary backlash against Greek influences. Funnily, Rome was allied with Carthage against Pyrrhus, which could mean a Carthaginian-influenced Rome.