Zefram Cochrane and Jean-Luc Picard should’ve met
Keep this in mind, Jean-Luc Picard was a history fan but never once had a conversation with Zefram Cochrane. He never even met the legendary pilot. For someone who is so in love with history and the past, you’d have thought the writers would’ve given Picard a chance to show off not just his historical knowledge but his appreciation for all the things Cochrane has done.
More of Cochrane dealing with aliens
We got to see Zefram Cochrane shake the hand of a Vulcan and teach them to dance. That was a hoot. It would’ve been even more interesting to see more of how Cochrane dealt with the very boring and bland alien race, as well as what he would’ve done if he had to face down a Borg drone. Cochrane was drunk most of the movie, and a lot of his scenes were a bit more humorous compared to the rest of the film. Seeing how he would’ve dealt with a Borg drone could’ve been great, even if it wasn’t played for laughs.
More of the post-war/pre-Federation Earth
While we got to see Montana in the post-war, we only got to see a small snippet and not a larger view of the world pre-Federation. While Star Trek has gone back in time before both before and after the events of First Contact, it has never really explored that time period. It remains an undefined and little-explored even. While showing the events of the fictitious Third World War wouldn’t have made sense in the context in the film, at least exploring it through dialogue would’ve been nice. It also would’ve been good to see more of the world as it was. Not just a small settlement in Montana.