Spock: "It reports that the jettison button was pressed before the Red Alert."
Shaw: "In other words, it reports that Captain Kirk was reacting to an extreme emergency that did not then exist."
But of course it does not report that. It only reports that Captain Kirk did not believe in an extreme emergency yet, and thus didn't press the Red Alert button to express such belief.
Button pushes are the thing being judged in the trial. But button pushes do not relate to emergencies. Button pushes only relate to Kirk's assessment on whether there exists an emergency or not. If Kirk doesn't press Red Alert in time, Shaw can only accuse him of failing to notice or express that there is an emergency there; the emergency itself cannot be talked away by the silly woman.
Sorry, my friend. Everyone in the courtroom agrees the jettisoning the pod would have been the acceptable reaction to an "extreme emergency" and the argument is that Kirk jettisoned the pod without there having yet been such a justifiable emergency. In any case, the jettisoning was assumed to be an emergency measure.
It makes some sense for Kirk to have the jettison button on his own control board. As it is such a dangerous post and the safety of the entire ship depends on the status of this pod, it seems reasonable that the captain has the eject/abort button at his sole use. After all a crewman's life is on the line and the decision to push the button would have to be made quickly. How does Kirk know when to push it? We never see the main bridge viewscreen. Presumably the required data was on display there. The dialogue suggests that Finney was in the pod
while it was taking its readings and remained there when it was jettisoned, only really he did get out in those precious few seconds and made it look otherwise.
I really just can't see it any other way.