Roddenberry pitched the following idea for every single Trek film but was shot down by some combination of Harold Livingston, Nick Meyer, Shatner, Leonard Nimoy, Harve Bennett, and so on:
The crew of the USS Enterprise travels to 1963 to stop the assassination of President John F. Kennedy. But it turns out that saving Jack Kennedy's life has dire consequences. In the end, it's none other than Spock who fires shots from the grassy knoll that take the world leader down.
You know, I love TVH, but since they are both time travel stories I must say if I saw the above pitch and a pitch for TVH side by side, I'd probably go with Gene's idea (as just ideas on paper). Who knows how it would have turned out as an actual script, much less a film. But I admit, the idea is very interesting to me. And it very much sounds like something that would have happened on TOS. Hey, it's better than Abe Lincoln appearing in space.
It does make me wonder though by the time they got around to TVH, did they say, "hey Gene, we love the time travel idea. But hey, instead of your JFK angle, how about we go back and get some whales instead?"