Well, it's safe to assume the series as happening in chronological order for the most part. I tend ignore stardates since they were nonsense back then, so The Omega Glory happened nearly a year before. Even so, there's no reason for Kirk to mention it to Flint. "Oh by the way, we were just a planet or so away last year. Hey, did you have anything to do with them having an American flag and the US Constitution? Hm? What's that Spock? Oh yes, whole crew. Dying. Never mind."
That's funny! As I understand it, "The Omega Glory" was originally written before STAR TREK's time setting was settled on, and the story was about humans from Earth who had colonized this distant planet maybe a thousand years prior to the episode. Once they put STAR TREK only 300 years in the future, "The Omega Glory" made a lot less sense / required a flabbergasting coincidence of identical documents.