Even though the museum is accessible only by appointment, it still actively adds to its collection and happily accepts donations (though only so long as they can supplied with appropriate documentation and have been correctly preserved) because they have potential research value in the future. The museum's collection was fundamental in discovering that DDT was affecting birds, for example.
Even though it's technically a museum, I think it can be helpful to think of it as an archive that can be used to pinpoint changes (to genetics, toxin exposures, early human technology, etc.) across time and across the landscape.
I think most major natural history museums will probably have large collections of specimens/artifacts that are intended strictly for research, not for the public to ever see. The small portion of the collection that is on display is fantastic fundraising, but public displays are not the only purpose (or even the main purpose) that a museum like the Smithsonian serves.
This is a very different use case from a baseball memorabilia collection, however, which has little value outside of being displayed to the public. I can see how such donations can end up being a white elephant gift.
Humans have created an absurd number of artifacts and what’s valuable isn’t the items but the time spent studying them. Just dumping a collection on someone isn’t doing them any favors, just handing them an obligation.