The concept of the shuttlecraft existed well before "The Enemy Within"; they just hadn't built the thing yet.
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Well if that's the case, why didn't they at least explain why the shuttle couldn't be used for a rescue?
Because they didn't know that devoted fans would be spending decades dissecting and analyzing every line in every episode of their show. And because continuity wasn't a priority back then; if anything, the producers of dramatic television shows tended to go for an anthology flavor even in shows with continuing casts. They didn't foreshadow future episodes or worry about overall consistency, because this was before large-scale syndication of reruns, before home video, before the Internet. An episode ran once or twice and then it was gone. So each episode was written and shot on the basis of what worked best for the episode itself, period, regardless of external considerations. And for "The Enemy Within," it worked better if the transporter was the only way to rescue the landing party. A reference to a shuttle wouldn't have served any story purpose as far as the episode itself was concerned; it would've only been a way of laying continuity groundwork for later episodes. But that just wasn't the way '60s TV producers thought.
