Most city states never grew past that stage because they limited participation to the citizens of the city. At some point these few thousand aristocratic families loose impetus and the decline set in.
Rome was rather a mongrel city from the start, picking up its people from a mix of Latin and Etruscan folks. As a small state in a region with larger ones, gaining people was a higher value than clean bloodlines. So when they got the upper hand against a neighbor, they would offer the aristocrats in that city a chance to be Romans, too. When Rome set up colony cities, this also gave a way for lower class Romans to become upper class in the new city, and then upper class in Rome with time.
This gave Rome a far larger population of loyal citizens to work with than, say Athens. The value of this was proved in the Punic Wars, when the Romans could lose battle after battle and not have the defections that Hannibal based his strategy on. Every city had a reason to stay loyal - they were part of the Roman core. If Hannibal one, they would be ruled from Carthage and have no such expectation.
This continued into the late Republic and Empire. The Spanish that the Romans fought in 80 BC were becoming Emperors by Hadrian's time. The Illyrian tribes that were pacified by Augustus were producing the Emperors that brought the Empire back from the 3rd Century Crisis. The Eastern Provinces and Africa also were happily producing the wealth that allowed the Empire to grow to Scotland.
Rome grew the most because it offered a way for all its peoples to help support it.