IMHO, said "elegant" concept was a humdrum pulp scifi design, whereas the flying brick we actually got at least had novelty on its side.
For the audiences of the 1960s, it would have been pretty easy to understand if our heroes had said that weather prevented the use of shuttles. Later audience generations tend to think the shuttles should be capable of weathering a hurricane, yet Trek is fairly consistent on the shuttles actually being very fragile and unable to cope with bad weather.
Should the episode have featured an explicit reference to bad weather (say, those high winds that seemed to tear clouds to shreds above the campsite)? Possibly. But the episode did throw a veritable hail of rescue ideas at the audience, only to debunk them one by one (either because there was a tech reason for them not working, or because the indecisiveness of the "good" Kirk made the ideas outdated by the time they might have gotten a clearance). We basically got the impression that our heroes know their stuff, that they know what is possible and what is not. If they didn't use the shuttles, then, it was probably for a good reason.
Timo Saloniemi