What you're missing is a history of conquest. Greece and Rome had a very intertwined cultural exchange. It wasn't just alexander that conquered land all over (all the way to india) and I am sure you know of how rome invaded it's way to the point they needed two emperors.
Civilizationd in america invaded each other but not the rest if the world. Same with Africa (outside of Egypt and few minor regionally invading countries). Even in asia, chinese never invaded india. Now Genghis obviously did more invading than alexander so I'd be interested in how much culture and history he spread but I think Rome is what made a difference in the western part of the old world. They built roads and encouraged people from different parts of the empire to travel and trade. Genghis had a habit of killing everyone that resisted but the romans kept them alive as slaves.
American culture and literature for example, I am sure will last many centtiries in other cultures.