. I was also thinking of the Bozeman, but the circumstances of their arrival is so damn confusing...and seems to be a predestination one as the universe doesn't change ala' "Yesterday's Enterprise"...sending them back would probably be too dangerous.
Who says the Bozeman
wasn't sent back eventually?
On the other hand, why bother? People traveling to the future aren't changing the past in the general case; nobody will miss them if they don't return, either because the future they end up in is the one in which they originally went missing, or because time "self-heals" minor things like that. And changing the past is the only thing our heroes ever worry about - insofar as changing it affects the quality of life in the present.
That way, only certain knowledgeable people like Spock or Scotty could modify the BOP engines to do the same thing while other ships don't have this ability.
So the mission to observe Earth in the 1960s was so super-secret that not even all that many in Starfleet were told of it?
Sounds a bit dubious, when we think how unimportant that mission appeared. Our heroes didn't exactly achieve anything they could plausibly have set out to achieve - they just had random adventures in the past. Why study the 1960s? There must be mysteries a thousand times more interesting, in time periods where fewer things can go horribly wrong. Say, before
the invention of high-power radars, or optical telescopes...
The episode makes it sound as if time travel is mundane and technologically uninvolving, and is applied on trivial pursuits. Which is gonna backfire pretty soon. And this is as good an excuse as any for the cessation of such things: Starfleet knows it's stupid to keep doing it, but Starfleet also has dirt-cheap, surefire, available-to-all methods of going back in time to easily hunt down anybody who goes back in time against Starfleet's warnings!