The refit Enterprise is a beautiful design and hugely influential within the franchise, but it’s nowhere near as culturally significant as the original. The original series Enterprise is the symbol of Star Trek and one of the most iconic images in all of science fiction.
Why should it be a competition? The Smithsonian already has multiple Trek items in its collection, including the encased Enterprise pendant from "Catspaw" and an AMT Enterprise model kit donated by the retired Naval officer who assembled it (according to the book Inspired Enterprise, which we're discussing in this thread). Since when was a museum only allowed to have one object associated with a given artist or topic? Exploring how a cultural idea or an artist's body of work evolves over time is part of a museum's function.
And so is preserving historical artifacts, whether they're publicly displayed or not. The refit Enterprise is just sitting out in the open in a lobby, not protected from deterioration. Why should its preservation be dependent on whether it meets some threshold of "cultural significance"? That's not how museum staffs or historians think. They want to preserve as much of the past as they can, not just the parts someone arbitrarily says are more "important" than others. After all, those decisions are inevitably biased, and different generations and subcultures won't agree on what parts of a culture "deserve" to be remembered by history. Any part of the past ceasing to exist through neglect or deliberate destruction is a loss to history.