I haven't thought about this story point a great deal (because it's frankly extraneous to the story Gene Coon was telling), but if I had to come up with a headcanon reason for it, I'd say there was probably something happening elsewhere that required the presence of the Enterprise. I'm imagining something similar to what we saw at the beginning of TNG's "The Schizoid Man," where the Enterprise-D went off to deal with a medical emergency while Data, Worf, Troi, and Dr. Selar beamed down to tend to Ira Graves. But whatever the Enterprise was dealing with couldn't have been TOO serious, since Kirk, Spock, and McCoy were all on the shuttle to get Nancy Hedford.Does anyone possibly know the reason - besides the obvious writers-answer that it traps the characters into the story until they solve the puzzle and win their escape - why The Big 3 and Commissioner Nancy Hedford are traveling by shuttlecraft to the Enterprise; instead of having the Enterprise travel to the planet to pick up Hedford?
Personally, I'm more curious why Kirk picked McCoy, Scotty, and Uhura to beam down with him to negotiate dilithium mining rights with the Halkans in "Mirror, Mirror." (Although that's equally extraneous to the story Jerome Bixby was telling.)