Howzabout this: The whole aparatus consists of more than just the jettisonable pod. The part that the person is actually manning is a small room just inside the ship's hull
...Such a "room" would probably best be explained as simply the airlock or access corridor through which Finney reaches into the pod to get the measurements going. At the moment of pod jettison, three things could happen:
1) If the pod is big enough, Finney could be in there, and might be launched with the pod - a potentially survivable situation, but only if the pod doesn't quickly get fried by the storm.
2) If the pod doesn't allow Finney to squeeze all the way in, or if Finney times his exit wrong, he could be caught in the doorway, and be sucked into space through the hole left by the departure of the pod, or perhaps cut in half by the closing door.
3) Finney might also manage to get out of the doorway in time, but be hurt somehow, perhaps by the door mechanism.
I agree that Kirk would be too busy to worry about Finney's fate right away. And (as per Kirk's damage report at SB11) there might be pressure alerts going off all across the ship at the time, so the presence or absence of one at the pod socket and the adjoining interior spaces wouldn't be an immediate telltale, either.
I guess its like getting a shot, or taking a band aid off. Do it faster, and it hurts less.
This sort of "nautical" reacting to spatial storms is also seen in ST6:TUC, where Sulu thinks his ship will better survive a tidal wave in space by turning her bow into that wave...