I do think a big reason for CBS going forward with a lawsuit now after years of seemingly not caring about "fan films" is a combination of "fan films" like Axanar getting too big and CBS launching their own TV series soon. Put simply, they don't want the competition, especially from groups that don't have the right to use their IP. If they let big productions like Axanar go forward, it will cut into their profit when they launch the new TV series since some fans may stick with the "fan films" and not watch the new CBS show.
Paramount did the same thing back in the 90s when they launched startrek.com. A lot of fansites and communities (remember this was just at the beginning of the 'net being a thing) found themselves no the wrong side of a take down notice. Don't recall anyone getting sued over it. Lucasarts did sue some people when they launched starwars.com--mainly to recover Star Wars related domain names.
So yeah, this could be totally about CBS seeing their market fora web-based Star Trek series being fucked over by fan productions poaching CBS's turf.
Think about this one: had CBS/Viacom moved Enterprise to the web like some of the Save Enterprise people wanted, this chain of events would have come about a lot sooner and most--if not all--of the current crop of fan productions wouldn't exist.