Are you sure it doesn't actually come from the word "pusillanimous"?
When a guy calls his bro a "pussy" he is not calling him a slang term for a fucking 16th century word no one knows. Nor is he calling him a cat because apparently cats have some notoriety for being cowardly? In the context of modern society it is incredibly self-evident what someone means when they call someone a "pussy" and it is so obviously a gendered insult I cannot believe people still bring this up. Also as if you really needed more evidence against it: pusillanimous is an adjective and pussy is used as a noun in every case I've ever heard it. That seems a little odd if it were shorthand, doesn't it? It's always said, "that guy is a pussy" not, "that guy is pussy". I realize this is coming off a little sharply, but you would not believe how much I see this used as a dismissal of a bigger problem.
That definitely could be one of the origins of the word's current use. . . along with several others. While pusillanimous is an excellent word meaning timid, it originates in the Latin word pussilus meaning very small. When a guy is called a pussy, he is being called very small. Given the word has so many meanings including kittenish, nowaypablo's original use of the word "weak" unambiguously expressed his meaning.