There is a DC Comics story set in the movie era, "Echoes of Yesterday" in Star Trek Special #3, in which aliens use the slingshot effect to go back in time -- but, conveniently, they learn how to do it by forcibly reading Kirk's mind. It's not 100% consistent with FH because Kirk says in the story that he doesn't know the details of how it's done, but with a little fudging, it can be interpreted that they extracted the special knowledge that Kirk, Spock, and Scotty kept to themselves according to FH.
So you're proposing that Kirk & company discover a means to travel back in time, keep important details from Starfleet and use that as leverage to get the mission back to 1968? What happens if they are lost on this or any other mission? The federation would have lost that knowledge forever. Sorry, doen't make sense to me.
If they kept the knowledge to themselves and NEVER used it again I could see your point but they travelled back in time using the slingshot maneuver just a few months after discovering it. Not to save Earth. Not to put things right. For simple historical research. What other information would the decide to keep to themselves?
They decided to keep Cochrane's existence secret because the galaxy already thought he was dead. And, as we saw in Federation, that came back to bite them pretty hard on the ass not much later. At least they had a good reason for that.
Sorry, don't buy it.
I think you should read Forgotten History, the novel explains why time travel seemed to focus on the Enterprise and her command crew and not as common as one would expect with the experiences they learned.
Besides, since you are a fan of Christopher's other works, if you buy this novel, it will help grow his royalty check so he can continue to write novels that we love.