The Romans took the ideas of the Greeks and made their own special addition, lex, i.e. the rule of law. This proved to be a culture superior to even that of the Greeks, which had become somewhat decadent by the time of Rome's ascendance. Rome eventually pacified and incorporated Greece, which then became a client state of Rome, a nice place to send the kids for college. Still, Rome's victory at its core was a Greek victory because Rome's dominance over Greece was born out of how well they incorporated the ideas of the Greeks and improved upon them. In essence, they had become bigger believers in the Greek approach than the Greeks themselves.
For a variety of reasons, Rome in its time fell, in part because of decadence.
Hanson is not really talking about Greece as a place as much as a state of mind and an approach to knowledge, and the Americans are the most successful society because they have most successfully adopted and adapted this approach in modern times, because it was transmitted to us through the classical culture of the British Empire. (Think of the lyrics of "Modern Major General" by Gilbert and Sullivan. If you were going to be an officer in the British military, it was assumed that you could read Latin and Greek.)
Think of it this way. If a modern day American scholar went back in time to Greece and used the words drama, comedy, democracy, tyranny, legal, idiot, senate, republic, empire, liberty, science, etc., the ancient speaker would understand because they would not be just understood concepts, but the words themselves are if fact the exact same words as 2,000 years ago. However, if you use these words in a "modern" Islamic society, they really won't know what you're talking about although they sometimes do a good job of pretending they do.
"liberty", etc are not the same words as used in ancient Greek, but your points are well taken and very well put.
In the 4th century AD they moved the capital of their empire to Greece (Constantinople), accepted Christianity in the interpretation of Greek Fathers of the Church and in the end adopted Greek as their primary language.
This new Christian Rome lasted another thousand years while in the West the "Rome has fallen".
Most excellent post bump.