"He, himself, was a secular humanist and made it well-known to writers of STAR TREK and STAR TREK: THE NEXT GENERATION that religion and superstition and mystical thinking were not to be part of his universe. On Roddenberry’s future Earth, everyone is an atheist. And that world is the better for it. — Brannon Braga, International Atheist Conference in Reykjavik Iceland June 24 & 25, 2006"
I'd agree that in the 1960’s the original Star Trek series’ atheistic plots were indeed more subtle than in the subsequent 24’th-century shows. They mostly involved the stories in which advanced aliens or other lifeforms would assert themselves as false gods, only to be debunked by the crew of the Enterprise. Such examples are the episodes “Return of the Archons”, “A Taste of Armageddon”, “Catspaw”, “The Apple”, “Who Mourns for Adonais”, “And the Children Shall Lead”, “Plato’s Stepchildren”, and “The Squire of Gothos”. The idea behind these plots is surely to promote skepticism of religious beliefs?
I do like to play devils advocate, so let's try that.
There are some episodes that show religion in TOS, notably the 'son of god' plot twist in Bread and Circuses. I also remember some character wearing a Bindi - a forehead marking, which I think is Hindu? I guess you could also argue that the altar in Balance of Terror is a religious symbol? Other than that, I'm coming up blank.