I think the story of how Zefram Cochrane salvaged an old USAF Titan II nuclear missile in its silo and turned it into a SpaceX-style SSTO stack with deployable warp nacelles is a more interesting story than the near-end of that story (the launch) we see in the film. Even the return to Montana would be interesting on its own. No, maybe we don't need to know the whole history of the Phoenix project for the film, but there's clearly an interesting backstory and afterstory to the film that would be interesting in its own right.
It's not impossible -- Tom Hanks (yes, that Tom Hanks) wrote a short story a few years ago, "Alan Bean Plus Four," about a group of Home Depot employees who build their own rocket for a four-person lunar flyby that struck me as sounding plausible.