What do you mean by 'Hollywood'?
Hollywood makes movies. Traditional tv broadcast channels (NBC, CBS, ABC and FOX) make television shows. Some of the traditional tv broadcast channels are now a part of the Hollywood studios (CBS/Paramount for instance) but they aren't considered to be 'Hollywood' and neither are standalone companies such as Netflix.
Paramount makes several Star Trek shows: Discovery, Picard, Lower Decks, Strange New Worlds, Prodigy. The second season of HALO started filming in September 2022 as well.
Disney makes Andor and the Mandalorian, Obi-Wan, and the Book of Boba Fett with a few other SW shows in production, such as Ahsoka.
Stargate and Babylon 5 are old shows. Other than a handful of cast members, reboots will require new actors. There has been talk of rebooting both, or some sort of continuation.
Then you have copyright. Who owns it and are they willing to allow the franchise to be messed with in a serial format. Some shows don't ever get made because the copyright owners can't agree.