Oh, and - religious skepticism wasn’t a crime punishable by death among the Greeks either. Every philosopher had his own idea of what divinity was, from ‘the limitless’ (Anaximander) to anthromorphic because we create gods in our own image (Xenophanes) to completely disinterested in human existence (the Epicureans).
How about Socrates “introducing new demons”? 
Socrates was deeply enmeshed in a sticky political situation, and the people in power were forbidden from trying anyone for political reasons due to an amnesty. Socrates was a pretty notorious intellectual troublemaker, and he continued to stir people up right to his death.
Besides, the charge of introducing new gods is clearly not a charge of atheism. It’s obvious that, historically, there were tons of thinkers playing with atheism, agnosticism and redefinition of the idea of the divine, and also that the introduction of new gods happened all the time.
So they wanted him out of their way and set up a kangaroo trial?
You know there had to be someone at the top of the mountain waiting for suckers to come along: “Yeah, I’m Zeus, bring me two lambs or I’ll smite you.”
I wouldn’t say that’s it exactly - he had a hand in educating two of the worst ‘traitors’ to Athenian democracy of the time, Critias and Alcibiades. Socrates didn’t quite approve of democracy, so in some senses he really was corrupting the young men - his students became fans of oligarchy.
You can also distinguish between philosophical beliefs about the gods from religious practice in ancient times. People were clearly free to think whatever they wanted about the gods, but someone who refused to participate in civic rituals would have been considered suspicious. I actually don’t know all of the details of the religious charge against Socrates, so maybe that’s part of it, but I can tell you certainly that there were many other philosophers with radical ideas about the gods that didn’t get any scorn or punishment.
That raises the issue of which came first: Olympus, home of the Gods, or Olympus, the big mountain that’s on the other side of the neighbouring village. Probably a lot of ancient Greeks would have told you “Sure, that’s Mount Olympus but it’s not the Mount Olympus.” A quick search of Google Maps shows there are at least ten Edens in the United States but nobody goes to them looking for the Tree of Knowledge. Like the Greeks, we understand that these local places were just named after the original divine one.
Well, a guy named Barzai the Wise did try to see the gods of Earth by climbing the peak of Hatheg-Kla, but he found out too late that the gods of Earth are protected by the Other Gods…
Religion in general in the ancient world was something rather different than what most people have today. It was often a matter of ritual and mythology rather than belief. Indeed, the question of “Are the Gods really on Mount Olympus?” was probably not one they would ask. Then they had philisophy on the side, which was a rather different thing. There were attempts to combine the two, but they didn’t get very far.