This CBS/Paramount 'rights battle' thing is probably the most enduring myth in fandom since "beam me up Scotty". Paramount made movies on licence from CBS to use their IP. CBS retain any and all rights to all Star Trek property except the right to make theatrical movies. Paramount could use anything they wanted in their movies (and did, including Spock Prime, a character directly from CBS's well worn sandpit). They chose to do a reboot/alternate timeline because they wanted to do Kirk/Spock/the 1701 in their own way in a modern production while providing a continuity handwave so it wasn't just different without explanation (they guessed, rightly, that Trek fans hate that). That's it. CBS retain ownership of all the Trek elements they used to make those movies, and could use them themselves if they wanted (have Robau show up, for example, or have Quinto play his Spock).
While "CBS/Paramount fight for Trek rights" obviously is bunk, I
do think the theory that there were some rights issued involved has some merits.
Remember: JJ. Abrams Star Trek (2009) was not just intended as a feature length movie. The original intention was to reboot
the entire franchise. With new spin-off television series, tie-in comics, books and everything else. JJ. Abrams has his own television company (Bad Robot), and Star Trek's IP was seriously damaged at the time.
The more plausible theory is, that Bad Robot had an agreement with
both Paramount
and CBS: They make Star Trek viable again. But, for the duration of JJ Abrams "original" new trilogy, if they wanted to make a new television show, Bad Robot would need to be involved. That wouldn't have been a bad deal for CBS at the time: Either it's successfull, and they make money. Or it isn't, then there isn't much lost (they weren't going to do a new Trek series at that time directly after ENT anyway).
But then, they didn't came to terms on how to proceed with the franchise (there were some rumors at the time about JJ Abrams being unhappy with CBS - with whom he
normally shouldn't have much to do if he only ever wanted to do a
movie, and not a television franchise). Then, the comic-book-boom took over television, leading to much more genre shows and renewed interest in a new Star Trek show, but JJ Abrams changed gears to instead do the next Star Wars. Thus, CBS just waited for the "time-out" of their contract to do a new Trek television series. The series was announced to shortly premiere after Star Trek: Beyond, but still in the 50th anniversary of Star Trek, where it would have made much more sense to release it. But then
Beyond was pushed back, and inexplicitly, the launch of a "new" Trek series was delayed as well, to
after the release of "Beyond". That happened way back before they even hired Fuller in the first place, so there were no creative reasons to push back a new Trek series. Which smelled like contractual obligations (You are allowed to release a new tv-series, but not as a "direct competition" to the current movie trilogy - if your not working with us on this).
Again: This is just a theory. But one backed up by some press-releases at the time ("we pushed the new series back, to avoid confusion with the movies"). Of course it's not the black and white Paramount vs. CBS theory that has run amok. More akin to MARVEL renting out their characters (FOX had the movie rights to Daredevil, as long as they continued to produce movies, so Marvel just waited until the rights to Daredevil reverted back to them from FOX, and immediately put out a new television series).
I
sure as hell don't know whether it's true or not. But I think it has at least
some merits, as it explains some shifting of release dates (mostly
away from the 50th anniversary, which wouldn't make much sense, giving free advertising and stuff, if there weren't other reasons involved) and lines up neatly with the rumors at the times and the chronology of what happened.