Aztecs not being able to use barbarians as builders greatly limits their potential snowballing. They have to actively defeat city states and other players to get ahead, I'd say that's a good trade off.
Japan's district bonanza sounds crazy powerful, but IIRC you can only build new districts every two pop, right? That kind of limits its snowballing effects, though it leaves Japan quite powerful later on.
America is like England, in that they don't seem to have anything especially snowballey.
I'd agree that Egypt also seems rather powerful early on, the whole river bonus for production looks quite useful, and the Sphinxes allow for you to snowball faith/culture if it's possible to build them early, but I'd still say China's Extra Builder Charge/Eureka/Early Wonder Rush seems way way more powerful.
I don't think they need to defeat the city state. They just need to stay at war with one. If the AI is as brain dead as the Civilization 5 1UPT AI, they'll be sending a steady stream of units that can easily be picked off. That will ensure a steady stream of builders for the Aztecs.
Hopefully the AI is better this go around.