I also am not well versed in ancient Greek culture. I tend to think the "eternal" Virtues of the ancient Greeks were the product of Socrates, Plato, and the other schools envisioned as eternal forms to guide public life and social structure. The focus on their Virtues would result in a social structure with guard rails designed to preserve what they believed was the unique status of being Greek.
On the other side of it, I tend to think of Epicurus, who preferred to avoid being a public or political person who thought more of virtue as subjective tools designed to improve the individual. Designed to perfect the individual's goal of happiness, rather than perfecting the social welfare of the city states and their colonies.
Am I close on this perspective or missing the mark by a mile?