The above highlights the dramatic/logical problems of the story rather nicely. If Finney went to the pod completely at random, because he was part of a duty roster 430 names long, then we're watching a coincidence of cosmic scale unfold. How often would the ship face ion storms? Not another time in the three or five years we witnessed! Yet Finney based his whole plan, calling for careful preparation, on such an event...
Of course Finney could and probably would have manipulated the roster. Yet it would help if the list of names weren't that long - if indeed only certain top officers had the training or the authority to operate the pod.
Another dramatic/logical shortcoming is the idea that a tiny pod would endanger the mighty starship in any fashion. It just doesn't sound intuitively possible; nothing else did, at least not in TOS. Unless we count the magnetic containment pod in "That Which Survives"...
Which makes me wonder if the ion pod in fact was unrelated to the ion storm and was not an observation instrument at all. Perhaps it was a key component of the ship's engines (or perhaps some other system Finney was in charge of), and was in danger of exploding in an ion storm unless carefully monitored? Kirk would want to keep the pod aboard as long as possible, because the ship needed the pod - but would have to eject it if problems arose. And Finney would be sent to establish that the pod was safe; if he didn't confirm it was, by taking the required "readings of the plates", it would have to be jettisoned at red alert by default.
I'm still more partial to the idea that the pod was a delicate scientific instrument that posed no danger and was not intended to be crewed, but was to be prepared for launch by a member of the crew and then fired away uncrewed to study an ion storm. In this analogy to tornado hunting, the preparations would have to be performed on short notice, since ion storms were unpredictable, and would call for a trained specialist. And like StarryEyed
suggested, it would be a rarely used and largely experimental piece of hardware, so the list of competent specialists would be short, but their workload high.
There'd be no scientific return if the pod were released prematurely (it would not be ideally positioned, and Finney would not have all the experiments running yet), but there'd be risk to the ship if she stayed in the storm too long - so Kirk would personally control the complicated joint action of the launch and the escape from the storm with a single button push. When red alert sounded, the crew would snap to desired action: Sulu would start saving the ship, as always in red alerts, while Finney would know that the alert was his signal to leave the pod prior to its launch.
Tornado hunting is a simple enough analogy here: it explains Kirk's seeming desire to sail into the storm and covers all the technicalities that follow with the same sort of accuracy we saw in Twister
- that is, sufficient for carrying the drama. Alas, it's probably not what the writer had in mind, so he didn't insert cues that would make it more obvious that this is a logical motivation for our heroes and villains...