It's difficult to know specifically what ordinary folks believed, let alone the literati. But as you said, even today people mix religion and politics even when they at least superficially understand them as distinct and separable. I think starting from the assumption that people today, at a fundamental level, think and behave the same as people from millennia ago, is a good starting point, just as it's a good starting point when understanding different cultures and ethnic groups today. When we depart from that initial assumption, even with good intentions (e.g. some other culture is more rational, etc), prejudice rapidly creeps in. But it's a choice, nonetheless, and can lead in different directions.
AFAIU, many historians believe, at least tacitly, that atheism wasn't a thing in the ancient world, and therefore that religious and mystical ideas were unconsciously and hopelessly intertwined and melded with other knowledge and beliefs, at least much more than today (assuming they even admit we still do it today). Probably because they understand atheism, and implicitly agnosticism and religious skepticism, as a modern ideology; which, as an "ideology", it is, but that's skipping ahead a few steps. They look for evidence to refute that assumption, and it's relatively scant (though not non-existent), for all the reasons most of history is lost to us. But if you start from the opposite assumption, that people think and behave similarly, I think the evidence strongly supports that the same intellectual dynamics were at play, certainly among the learned. Emphases and perspectives are different--even today each generation is more interested in certain questions than others. And of course literacy and, presumably, exposure to diverse ideas was less common (that was my point about wealth). But AFAICT and IMHO all the same threads are there, the distribution is just different. If you were teleported to 100 BC, I'm confident you could find people with very modern ideas and perspectives, they'd just might be more difficult to locate. But I think even the general milieu wouldn't be too foreign, depending on time and place. (If you teleported to the US during one of the Great Awakenings, the milieu would be much more religious than at other earlier and later times.)
What definitely stands out in the surviving works is that atheism was generally cast in a negative light. While there's often open derision of magic and aspects of foreign and cult religions, religion was generally understood as an important element of a healthy polity. Though, in Plato's works its arguably (IMO conspicuously) ambiguous whether sincere belief is necessary, or just tacit acceptance and active participation in rituals. But none of that is unlike the situation until 50-100 years ago in the modern Western world.
Anyhow, among the few books that directly speak to this topic are Atheism in Pagan Antiquity (1922), https://www.gutenberg.org/ebooks/28312, and Atheism at the Agora: A History of Unbelief in Ancient Greek Polytheism (2023). They contain lots of references and quotations to classic works. The first book I came across via Reddit: https://www.reddit.com/r/AskHistorians/comments/9gtbs6/comme...