Wasn't it intended that the logs are recorded after the events, but in a dramatic fashion?
All of TOS was a holodeck re-creation being watched by Riker and Troi. The Captain's Logs were Kirk's recollections of events that were used to create the holo-simulation.
More seriously though, I see the logs as a plot device akin to the voice-over narration of the 1950 film
Sunset Boulevard. For people who are aware of that film, the film start's with William Holden's character (Joe Gillis) floating dead in a pool...but then we begin to hear a voice-over narration from Joe Gillis, recounting the story of how he died.
Granted, Kirk isn't dead when he is providing the voiceover narration of his log, but it's also true that he isn't recording his Log while we are seeing the Log events unfold, either -- but rather something akin to an omniscient narrator. In neither case, I don't think we are really supposed to take it at actual face value that (1) Dead guy Joe Gillis is
really able to tell us why he's dead, and (2)when Kirk is busy fighting Gorns and stuff, he
really isn't giving us a step by step recording of that fight.
It's just a plot device and shouldn't be taken so literally, just like people watching
Sunset Boulevard should not stress too much over "how is the dead guy telling us the story?