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?
Is this surprising? Classified missions happen all the time in the military. Any potentially dangerous information is probably kept on a need-to-know basis. Probably quite a few missions we saw in TOS were classified. "The Enterprise Incident," obviously. No doubt the orders to go to Organia in "Errand of Mercy" were classified so that the Klingons wouldn't get word of it. Same for other overtly military missions like "Balance of Terror" -- which has the added issue of the Romulans' relationship to the Vulcans, which was probably kept secret at first out of fear of racial backlash against the Vulcans (and which, according to some tie-ins, had been known as far back as the Earth-Romulan War but kept secret ever since for just that reason).
Given the dangers that time travel poses, it's just common sense that the details of how to travel in time, or the location of something like the Guardian of Forever, would be highly classified. Certainly a free society couldn't and wouldn't restrict theoretical research into temporal physics, but there could be legal restrictions on actually attempting time travel or possessing a time machine -- analogously to how there's no US law against studying how to build a nuclear bomb or publishing the instructions, but it is highly illegal to possess unlicensed nuclear material or explosives in general.