Indeed, it takes some doing for the audience to side with Kirk in this episode... The writers were clever enough to set this up as a rendezvous deadline, meaning that Kirk would win nothing by reaching his destination too early, and could thus at least initially defend his weird decision to stop for sightseeing. But it does make one wonder: if the crisis really were that severe, why could the Enterprise
not take the medicine all the way?
I mean, we can postulate all sorts of good reasons why that other ship could not arrive earlier, or could not meet Kirk closer to Murasaki, because we know nothing about that ship or her mission. But we do know about Kirk's...
The only reason I can suggest is that Makus III is already relatively far away, yet the New Paris colonies are at the other edge of the universe, and Kirk won't be able to go that far, despite the fact that he often takes multi-hundred-lightyear side tours on his own account; the scheduled supply run is the only one Starfleet can afford to send for the multi-month trip, from Makus III onwards. But if New Paris is that far away, travel time -wise, then why not delay the departure of the supply ship by a few days, which represent only a fraction of the total time and won't mean much to anybody?