I was trying to speak in a more general way.
Today what phenomenon can we observe to argument in favor of the existence of God? Nothing, the only mystery here is the origin of the creation of the universe. Which is not really a "daily phenomenon".
Thousands of years ago, everyday phenomenons could be explained only by the existence of gods. How could a woman give birth? Gods of fertility. How can the sun shine so bright and go throw the sky? Gods bringing a huge fireball. How storms lightning and volcanic eruptions appear? Gods Gods and Gods.
Coincidences, accidents, etc happen daily and religious people interpret them as acts of god. Many still don't believe in evolution. So hold the variety of life they see as proof of god. Others believe the origin of the universe can never be understood. So more proof of god for them. We know differently. But it's their choice to believe that. It's their faith. Maybe if we go back far enough there were cultures that didn't have any other explanations except gods like you say. So maybe faith wasn't as needed to believe in the supernatural because there weren't competing explanations for most things. But we know religion didn't play a big part in all cultures. Some had almost no rituals. Some no gods. So faith didn't have to play a big role in daily life for everyone. But cultures where religion and faith played a big part in people's lives it was their choice that it did.
Of course some had great explanations, it was the case for the most cultivated individuals and lots of scientists and philosophers were probably atheists. But it depends a lot of the period of time you're talking about (because Greek kingdom and Roman Empire cover a long period) and which part of the population.
From the beginning of the period you said, ancient greece, they already had lots of alternative explanations for what they saw in daily life. Atomism like I said for example. That's back in the 5th century bc. India had something similar to atomism around that time too. Before that there were explanations for natural phenomena like lightning, earthquakes, the sun, etc without gods. Maybe those ideas weren't more talked about because of connections to atheism and the risk of death or exile. Or maybe it was more difficult to spread those ideas because they didn't have the practical applications they have today. So it didn't matter much if gods were doing things or if it was something natural. But choosing to believe the more complicated explanation of gods doing things needs faith.