For what this datapoint may be worth, in my young years of watching Star Trek (which would be late 60s) I too found the lack of continuity annoying, not only in it, but in other dramatic series. I disliked the way relatives and old friends appeared from nowhere in order to die or betray or be thrown into plot-inducing peril, then vanished completely. I disliked the way astounding scientific discoveries would be made in one episode, then fail to change anything at all in the following. And so on.
I knew it was so the episodes could be shown out of order in syndication. That did not console me.