So, what direction do you think things would've gone in for Kirk, Spock, McCoy and the rest if Desilu had made a sequel series, or maybe some movies or something, after the 1966 series ended? If Shatner hadn't immediately gotten that job playing Denny Crane on that huge spin-off sequel to Perry Mason, and Nimoy's singing career hadn't taken off big time like it did (more number 1s than the Beatles and Elvis combined!), it seems to me like there were still a lot of stories to tell.
Star Trek Phase II. You did know about that, didn't you? Granted it would have been produced by a new fourth network under Paramount and not Desilu, but still... Star Treks I, II, III, IV, V, and VI? Or do you mean immediately after 1966? Well, I doubt that would have happened, since nobody knew how popular Trek would become until several years later. But if Desilu had produced a sequel immediately after TOS ended, I doubt the storytelling would have been any better than TOS's 3rd season; it would possibly have been worse.
Desilu was absorbed by Paramount in 1967, during Star Trek's second season. NBC wanted to revive Star Trek as early as 1972 but decided to make an animated series instead when they didn't want to invest in the budget into a live-action series. '70s TV strikes me as very cheap and it wasn't a good decade for science-fiction on television... ... but anyway, we'd get a second TOS series a year later, in 1973, instead of TAS if they did invest the money and Star Trek would've been updated for super-funky '70s. TOS attracted a lot of teens and young adults, so they'd want to make the new series look hip and forward. Just imagine what that would've looked like... or maybe you don't want to. I kind of like the '70s asthetic. You'd see a lot less Cold War allegories if any at all. Mission: Impossible was the same way. After their fourth season, they moved away from made-up Soviet-like countries and the IM Force fought the mob instead. On the movie end, Gene Roddenberry was interested in doing a prequel to TOS showing how the crew met and got together. He had the idea before Harve Bennett or JJ Abrams. This was either during the late-'60s or early-'70s. I don't know which.