Although the reason all of this happened is unclear, IMO, it seems to be a culmination of human social evolution.
James Burke’s series “Connections” often linked how it took specific developments in science and technology before other advances in science and technology could happen. An example would be a digital watch: it requires specific advances in plastic, batteries, the IC chip itself, etc before it could be developed.
Likewise, I think the social advances in the Axial Age wouldn’t have happened without the previous development of human society, literacy and technology to sustain a city such as granaries and aqueducts.
Such developments would spread relatively quickly due to the human propensity to control their environment, including making war on their fellow humans. Such a city-state would have a distinct advantage over individual villages and nomadic tribes. Conflicts would occur and the city-states have the advantage.