First off, does everyone realize that CBS, not Paramount, has the TV rights? Paramount has nothing to do with this.
Abrams' has a strong rep in TV and film, of course. But whether or not he, or anyone else, wants to do a TV series isn't the biggest problem.
The biggest problem is a business problem, not a creative problem: that CBS has the TV rights to Star Trek. They wouldn't produce a series for CBS - totally wrong demo. CW is wrong, too. Showtime might work, but would Showtime go for a franchise that is associated with free TV? Not the right image for them.
CBS could also produce a series for another outlet - I can see TNT or FX being excellent homes for Star Trek (not SyFy though!) - but is CBS motivated to make a series for channels that are not part of their corporate structure? Why not make shows suited to CBS/CW/Showtime instead?
But getting back to Showtime, here's an element we don't know, that could be a big factor. HBO has Game of Thrones; Starz has Torchwood. What % of viewers actually subscribed to the channels because they were pre-existing fans? That would be easy to deduce from the number of subscribers who sign up right before the series starts. If it's significant, that's a big deal, because pay cable survives by attracting and retaining subscribers.
Having a show that attracts new subscribers is much better than one that simply retains them. You can't find that number in the Nielsens, which is the only number we ever see. But if Showtime is savvy enough to ferret out this info on their own (which HBO and Starz certainly wouldn't volunteer, to them or to us), then they might see the value in following suit, by picking up a franchise with a large and loyal fanbase that might shell out $20/month to see something they can't get any place else. In these days of cord-cutting, that could be a big inducement.