I doubt Kirk would have bothered to check his own bridge cameras for verification that he pushed buttons in a specific order. He knew he had pushed them in the right order, after all. And the prosecution already opened its case by establishing that the computer had
indicated Kirk pushed those buttons in the wrong order; visual evidence would be superfluous (if dramatic) and showing it early on would really just make the court think that there was some reason to doubt the words of the prosecution.
I guess the more interesting question is why cameras around the ion pod were not checked. The 1960s audience might not have expected them to exist - and indeed TOS doesn't indicate that there would be cameras everywhere. Just remember how Scotty had to rig extra visual recording gear for the Martine/Tomlinson wedding... But the 21st century audience would certainly assume that the ion pod, or at least the corridors close to it, would be under visual surveillance. Now, Finney would definitely have made sure that surveillance didn't thwart his dastardly plan. But if the defense thought that the visuals on bridge activity were wrong or tampered with, but couldn't prove anything, they'd certainly go through the ion pod camera material with a fine comb as well, looking for discrepancies or oddities they could use for propping up their faltering case.
(Of course, there'd probably not have been any cameras in the pod itself, if we assume that no other remote technology was available, either. If Finney had to manually operate stuff there for treknological reason X, the only way to get visuals out would obviously have been by midget draft artists and carrier owls.)
Generally speaking, I'd think the whole ion pod affair would ring true to those who were familiar with 1960s or preceding military technology. Lone assignments where men risk their lives in servitude to stupid pieces of machinery were and are very much part of the military life, especially so in hostile environments such as the sea or the air. What might bother the audiences is that a high-ranking officer was lost, in a job that to their experience should have been assigned to an expendable enlisted.