"1 to the 4th power", and Jamie's cheesy-weird emotional rollercoaster, and Kirk being prosecuted by his ex, and Stone's inexplicable haste in accusing Kirk in the first place are all secondary and can be forgiven. Even the notion of an "ion pod" can be overlooked.
But the fulcrum that this episode rests on is not the computer. It's Ben Finney. How a clown like Finney could graduate from Starfleet Academy is beyond me. He concocts this whole scheme to discredit Kirk, and sabotages the ship as well?
If the courtroom drama seems cheesy, just remember, this is a depiction of a society where no captain of a starship-of-the-line had ever been put on trial before. (Also a society that, to date, had never had a mutiny occur aboard a starship, either) A society with that kind of record would be significantly different from ours.
Interesting how Kirk has to defend himself against the (altered) flight recorder record, but he has to sign a hardcopy report before his claims come under fire. (I don't remember ever seeing that in TOS again.) Usually, Kirk would simply sign one of those iPad/clipboard thingies.
Am I the only one who finds it improper that Kirk seems to repeatedly horse around with women he meets in professional situations where he should be expected to keep it, um, professional?
Despite all its flaws, "Court Martial" is a charming and entertaining "Kirk vs. Computer" tale, even if the real issue here is how infallible Starfleet ever managed to overlook a would-be mass-murdering terrorist like Finney in the first place.