No, I don't think the fans own Star Trek. But, @CaptGrumpy, if it wasn't for the fans, C/P wouldn't make any money from Star Trek. The fans are what drive Star Trek. No fans, no show. Star Trek is FOR the fans.
I don't think Star Trek is "for the fans" nor do I think that the fans "drive Star Trek." There just aren't enough fans out there to make Star Trek profitable when compared to the other major franchises in existence right now. Instead, C/P are interested in appealing to the broadest audience possible. Nevertheless, being a fan does not entitle someone to infringe on copyright nor does it excuse the behavior. C/P are very generous to allow fan works--particularly fan works that require crowdfunding campaigns. It's a good compromise between the official works that keep Star Trek profitable and fan works that allow fans to explore what they want to see. It's just too bad a small group of people had to try and exploit C/P's generosity and goodwill.