The role of the Christianity referenced in Bread and Circuses has to read in context.
The "Romans" in Bread and Circuses are us (US us,) complete with TV car commercials. And the episode says their empire is morally wrong. The Christians are there to appropriate the more or less compulsory approbation for Christianity on behalf of a critique of contemporary society. The supposed reveal at the end is just a shaggy god gosh, wow, cool moment. Like so many cool things it is actually kind of stupid, but there you are. My experience in life suggests that this kind of moral ju-jitsu never really works.
Historically "God" and even "Providence" have stood in for nature. You have to judge these things from context. Sorry a dictionary can't substitute for reading.
And, no, it is not an accident that Kirk never met a God figure he didn't reject. He's no Sisko.