Why would they put "Star Trek" on a cable network instead of their own network?
It is their "own" network - CBS owns Showtime and CW. It's common for corporations to own both broadcast and cable outlets, and makes sense because they are good at different things, have different business models and different audiences.
CBS shows need big ratings to compete - 10m or more.
Star Trek won't get that on TV. Plus the audience is totally the wrong demographic
.
CW shows get by with much smaller ratings, and I could actually envision a space opera that succeeds on that network, but the audience demographic would require an approach to
Star Trek that I don't think many of us want to see.

(I wouldn't write off a Melrose-Space type series out of hand, but I don't want to see the
Star Trek label on it.)
Showtime shows get by with smaller ratings still. There, success is measured by the ability of a show to attract new subscribers, but huge numbers aren't required, a million more subscribers would be just dandy. I think
Star Trek could pull that off without breaking a sweat (remember, I'm talking one million
subscribers, not simply people tuning into a channel they already have access to, which is a higher hurdle, but still.) And Showtime is the only one of the three that wouldn't mangle the content of
Star Trek into godawful forms to appeal to their audience.
However I rather they get their asses going on making movies.
That's Paramount's job. The "they" for the TV series is a different "they." No reason "they" both can do their thing, either in collaboration or completely independently (though it would make sense on both sides to collaborate enough that they don't cause needless audience confusion that might harm both sides.)