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.