Is this surprising? Classified missions happen all the time in the military.
The surprising thing about it is the apparent low priority of the 1960s mission. The heroes aren't explicitly about to do anything comparable to their other time travel hijinks - no saving the universe, no saving the Earth, even. There should
be a thousand time travel sorties more pressing than that. So the implication is that Starfleet is sending thousands of time travelers to do things ranging from the fantastic all the way down to this ho-hum observation mission. Which makes secrecy unlikely, and any uniqueness of the Enterprise
or its heroes in time travel ability implausible.
As for the need for secrecy, well, there does factually exist something of a gulf between those who really know the best, and the downright idiotic bulk of the population. It's just a question of whether the better-knowers are actually in charge of the secret things, or whether a sub-group of the idiots is... Democracy and openness doesn't help in that respect a bit.