I say: clean it up, and display it in the proud fashion that it deserves.
Well, first of all it's competing for space with the whole modern real world history of flight. This is one reason it's been moved around so much over the years.
At least in the glass case in the gift shop it seems to have found a long-term home where it can be displayed at eye level (it was briefly displayed this way on the museum floor after its return from a national tour in the 90s; other than that it's always been hung from the ceiling somewhere or another).
Notably, your other examples - Kermit and the ruby slippers - are part of the collection of the National Museum of American History, which is where other TV and movie exhibits are displayed and where many at the Smithsonian have always believed the Enterprise actually belongs.
And finally - fine, display it. How? I suppose I'd rather see it supported in the plexiglass truss arrangement than gutted and rebuilt. Everybody talks about how it's so important an "artifact" but it seems as if a lot of folks just view it as a sideshow exhibit - its preservation being less important than the opportunity to gawk at it. Skinning it on a more durable metal armature is not preservation by any stretch.