I agree with the timeline you lay out above, and also that Kirk initially believed that Finney had gotten out of the pod. And that he'd given Finney every opportunity to get out before jettisoning. The problem I have is that 'giving him every opportunity' isn't enough for an operation that's optional and can be aborted if necessary. Essential steps in such a process would be positive verification that the crewman was out before jettisoning, not hitting that launch button until you had such verification, and simply aborting the mission if that verification never came before it was necessary for Enterprise to depart. To my recollection there's no discussion of anything like that. Finney had plenty of time, but an accident or mishap could've occurred that would prevent him from getting out, and I'd think they'd have procedures in place to prevent the pod from launching with a crewman still inside.
Now it's possible those things occurred off-camera and Finney altered those records as well, but the problem I still have is that after all is said and done and it's discovered that Finney didn't escape, the investigation and court martial doesn't focus on "how & why did you launch the pod with a crewman still aboard when your (altered) computer records clearly show he was still inside" or "why did a needless death occur in a situation that was preventable and why weren't there sufficient safeguards to prevent something like this from happening?" Rather the entire investigation & proceedings for the remainder of the episode focus on "were you or were you not at Red Alert when you jettisoned the pod?"
It's this sole focus by the Commodore and the court on the Yellow Alert vs Red Alert condition that bothers me. If Finney's death was needless and fully preventable, then Kirk should've been facing an investigation & court martial as soon as he arrived at Starbase. Instead he doesn't get into trouble at all until the Commodore suddenly realizes the ships records don't show they were yet in a Red Alert situation, and it's that specific focus point that suddenly shifts the direction of the entire episode. Implying that something that was previously considered justifiable and simply unfortunate when at Red Alert, is suddenly unacceptable at only Yellow Alert. It should never have been acceptable under any Alert condition for a crewman to end up dead in an abortable operation, and Kirk should've been in hot water as soon as Finney was presumed dead.
To me, if the makers of the episode viewed the pod jettison as a standard part of the operation, then they would've written in at least a little something about Finney's death being unnecessary under any circumstances, and had Kirk question himself as to whether he could've done more to make sure Finney was indeed out beforehand, rather than focusing solely on when in the sequence of events did the death occur.
I'll admit I haven't watched the ep in awhile though, and will try to review tonight to see if I'm misremembering. Are episode transcripts available online anywhere? Just curious.