It shouldn't mean assume the accused party is guilty, it should however mean don't dismiss the injured party. If someone came up to you and said "James broke my leg" the response shouldn't be "No he didn't", but it also shouldn't be "Let's beat him up". It should be "Hey lets help you get that leg fixed up, and help you get an investigation started". What people presently face with sexual assault, especially when bringing up cases against people in power is "No, not them, that didn't happen, you're making that up because you want something." It's not limited to just women, Terry Crews was sexually assaulted in an elevator and couldn't even use his massive muscles to pop them like an orange because of how that would appear. He didn't speak out because he didn't think anyone would believe him. So when people say believe the victim it doesn't mean assume the accused party is guilty, but rather it means don't just dismiss the victim when they say someone hurt them.
What should guide the rhetoric of social movements is, to take it back to your college GE writing class, the consideration of "writer-based" vs. "reader-based" writing. In the former, you write what you feel based on how it makes sense to you; in the latter, the intent of the writing is to convey a message in the way that the audience can comprehend it.
That hashtag will and does imply "the accused is guilty." Something that would convey the intent you describe more directly would be along the lines of #DismissNoWomen or #TakeHerSeriously.
It does mean that. That's what the word "believe" means. I understand you're trying to say that people's intentions are to instead support all women, but that's exactly the problem. That's not what "believe all women" means.
All these shows won't turn out well for anyone under senatorial property qualifications (SPQR style, I mean).
Alas.