Why send a shuttlecraft full of people out on a non critical mission when you have a more important mission that should have taken priority? Deliver the drugs for the plague victims first then come back to investigate the quasar.
But Kirk doesn't have spare time. The very fact that nobody has visited Murasaki before suggests that the unusual shortcut Kirk took in order to deliver the medicine would be rare indeed. And since nobody indicates that Murasaki would be
news to Starfleet (it can no doubt be seen from a distance of hundreds of lightyears at least), the failure to investigate it at close range before this must be blamed on lack of resources.
The time window built into the medicine delivery mission is the only one available for studying Murasaki. That Kirk would do it before rather than after delivery is just his judgement call: health issues and science opportunities are both important, but can be simultaneously accommodated here. Perhaps Kirk has already been told to head to direction X when his time window closes, and doubling back to Murasaki would make that inconvenient?
Note also that Kirk is headed for a rendezvous, not a delivery. It makes no difference whether the medicine arrives in the first day of Kirk's time window, or the third - it won't continue to the end users until after the third.
Personally, I feel the writer handled these issues very well indeed, making the overall scenario plausible. The technobabble on why exactly the shuttle got stranded leaves a bit to be desired, but it's a fun thing to rationalize away. What the plot fails to properly establish is why there is hurry - the castaways are in no danger of perishing any time soon. The inability of Kirk to return to this location after his time window closes could no doubt be overruled if lives were at stake...
Sure, Kirk tries to justify it by saying he has standing orders to investigate quasars but he's the Captain, he easily could have made a decision to deliver the medical supplies first.
And conversely, he could easily make the decision to survey the quasarlet first, and so he did.
Also, what's the point in sending out 7 people to investigate? What exactly were they going to do?
It seems even Kirk didn't really know. There were four star systems that the team might have surveyed, and Kirk would probably head for the one at the #1 slot on their schedule when hearing of their disappearance - but Kirk indicates no familiarity with such a schedule.
What's the advantage of sending 7 people instead of a probe?
No need to assume they didn't send a probe. Probably they sent two dozen.
Plus a shuttle, for tasks probes weren't good at.
The one thing that did seem implicitly ruled out was sending the ship herself, either for the survey or the rescue job. Apparently, mini-quasars aren't good for the health of starships! Which also helps explain why Kirk ordered just warp 1 and not maximum warp when finally departing the region with the survivors on board.
Timo Saloniemi