Thanks for the vote of confidence.
It’s complicated, but the short answer is “yes” Whenever you discuss mythology, you’re really talking about someone else’s religion. This is true of mythology the world over, and ought to give you pause any time you think that someone else’s myths are silly or trivial. Remember, to them, your own religious beliefs can be cast in an equally unflattering light. (Being an atheist or agnostic doesn’t get you out of this. You still have some philosophy or beliefs behind your actions)
In the earliest days, I’m sure that the Greeks and Romans did believe their myths as certainly as any group does. It lay at the basis of their history and science and politics. What makes people question the Greek and Roman sincerity of belief is, I suspect, the sophistication of their culture and the ease with which, in later days, they modified or defied the beliefs.That seems like the behavior of someone who doesn’t really take the religion seriously anymore.
There is a section from the Phaedrus of Plato that I quote at the start of chapter 5 of Medusa. Socrates and Phaedrus are walking by the Ilissus River near Athens, and Phaedrus asks Socrates if they are near the spot where Boreas, the North Wind, is supposed to have abducted the maiden Oreithuia. The story is told or alluded to by many writers. Pausanias, who in the second century CE wrote a guidebook to Greece that still exists, points out the spot. Socrates says the spot is a little further down than they are at the time, and that there’s an altar to Boreas at the spot.
Phaedrus says: “But tell me, Socrates, before Zeus, do you believe this mythical tale to be true?”
Socrates responds that "if he disbelieved it, as the wise do, he wouldn’t be unusual. People have tried to account for this and other myths with rationalizations. This, says Socrates, requires a lot of leisure time, and he himself does not have much of that. To him, the truth or lack of it in the myth of Boreas is not important, and not worth arguing over. Real philosophy was worth arguing over, not speculating about the apparently fantastic with little information.
It seems to me that there was a range of belief in ancient society, with the sophisticates believing little or nothing, down to fundamentalist literalists. But everyone observed the forms and practices of religion – the ceremonies, the feast days, the household gods, and so forth. And there were a LOT of temples and religious festivals. If you don’t believe that, have a look at A.B. Cook’s massive three-volume (although it’s five physical books) work Zeus: A Study in Ancient Religion
And it’s not that different from today, when you have unbelieving atheists and agnostic, middle-of-the-road practicing communicants and dyed in the wool literal believers.
If you read the various writers on mythology you can identify their own philosophies and outlooks. Apollodorus, whose Library is a treasure trove of background, but who tries to cram in every known detail and keep it consistent. I think of him as a Continuity Geek, but for serious religion rather than, say comic books. Ovid, on the other hand, regarded the myths lightly, and had no compunctions about changing details or even inventing new versions if it suited him. Dionysius Skytobrachion was even more extreme, completely re-arranging the myths. But the core of the myths remained constant, repeated by countless other voices and represented in vase paintings and other art. The excursions of Ovid or Dinonysios didn’t make much headway against the inertia of belief.
So did the Greeks and Romans believe in their myths, or were they the “gay fantasies of the poets” as one saying had it? Overall, most people believed at least some of the myth, and acted that way. Those that didn’t still observed the outward signs, whatever they may have felt inside. I suspect that screenwriter Dalton Trumbo in Spartacus had it dead on when his invented character Gracchus purchasing a chicken to sacrifice to the gods “I thought you had reservations about the gods,” remarks Julius Caesar. “Privately I believe in none of them. Neither do you. Publicly I believe in them all.”