I think you may have unknowingly picked a pretty obscure time, early Mycenaean Greece, predating the more familiar classical period by about a thousand years. Maybe someone can weigh in on what's known about ancient Mycenae religion but assuming you're asking about religion in general in ancient Greece/Rome, this compilation of links by u/gynnis_scholasticus gives some insight: https://www.reddit.com/r/AskHistorians/comments/yag8kx/were_there_atheists_in_ancient_rome/itbuvwy

There's also an answer from u/JCurtisDrums

All those answers I think converge on a few points: the gods of the ancient world didn't really care or even necessarily knew what you believed (unlike the omnipotent god of modern Abrahamic religions), they just wanted their stuff, be it pretty temples, animal sacrifices, or words of adoration and rituals of worship. Could you be an atheist or agnostic (by our standards I think agnostic is the most you'll find)? Sure, but it didn't matter because unlike today (in many countries at least) religion was intertwined in day to day life; it was completely inseparable. There was no separation between the secular and the religious. You would perform the rituals, make your sacrifices, etc. or you wouldn't be a member of society in good standing. That's different from today (again, in many countries) where a person can ignore, violate, and eschew the teachings and laws of every religion they please as long as they follow the secular law which is (in theory!) separate.

Now all that said, that's not exactly your question. Some very educated people could take bold stances on the subject but you're asking about the common people on the street. I think the answer is that your common, "average" person is going to believe in gods of some sort. Ancient religion isn't really standardized, there's no bible with instructions for worshipping Zeus, or Odin, or Marduk, or whoever, so your own personal understanding might vary, but again that doesn't matter so much as long as you do the right things.

You ask if we overestimate religion in people's lives, I think the opposite is true. Fiction and popular history has to really downplay the role of religion and the gods in people's day to day lives or it'd be just too weird. One of the answers notes Cicero talking about the widespread belief in divination and oracles and the such. People, all the way up to kings, would be basing decisions on how a sheep liver looked, or the movement of birds or the movement of stars (it's easy for me to forget how insane the night sky is without modern light pollution), because that was how the gods could try and answer questions. I think with that complete immersion, from birth to death, it'd be pretty hard for a regular person to think too far outside the norms.