When Moe Green tells Michael that the "Corleone Family don't have that kind of muscle anymore....and they were being chased out of New York by the other families" what did he mean?
True, the Corleones were not at the peak of their power, and the other families were chiseling away at them a little. But there is no evidence to support the contention that they were being "chased out of New York."
But keep in mind what he says right after that statement. "I talked to Barzini." I think that's the source of his fatally wrong impression of the Corleone family.
And you know Moe, headstrong, talking loud, saying stupid things. Not smart enough to objectively evaluate the information Barzini is feeding him.