I put this down to the Greek belief that the world was fundamentally explainable by rational means. That very modern viewpoint was totally at odds with their own religion, and the deeper thinkers of the time were painfully aware of the paradox. Yet nevertheless, give a Classical Greek a problem, and he would instinctively look for a rational solution. This rather suits me as a mystery writer.
The unusual converse however, is that although people didn't believe the dead visited the living, they did believe the living could visit the dead. Greek mythology is full of people just popping down to Hades for a quick chat with the shade of someone long gone. Theseus, Heracles, Odysseus, Aeneas and Orpheus all take the plunge and return to tell the tale.
The important thing to a Greek was to make sure the spirits of the dead made it into Hades, after which they weren't coming back. This was largely arranged via the funeral ritual.
Here is Nicolaos, visiting the body of his first ever investigation:
I stepped forward to the body, as was required. An urn of ashes had been placed there. I dipped my hands in, raised them high above me, and poured a handful of ash over my head, felt the soft falling touch against my face, and the harsh burnt smell in my nose. Looking down I could see the pattern of black and white specks on the floor all about me, where every visitor before had done the same thing.
I cried and lamented for the shortest time I decently could, inspecting the body all the while. Ephialtes had been dressed in a white shroud. A honey cake rested by his right hand. A strip of linen had been tied around his chin to the top of his head, to keep his mouth shut, by which I knew the coin, an obol, had already been placed in his mouth.
Ephialtes would give the coin to Charon the Ferryman, who would carry him across the Acheron, the river of woe, on his way to Hades. He would cross the river Cocytus of lamentation, and the Phlegethon of fire, before coming to the river Lethe, where he would dip in his hand and drink of the waters, and so lose all memories of his earthly life, finally coming to the Styx, the river of hate, after which he would be in Hades, and remain there for all eternity.
You'll notice I managed to get in the basic geography of Hades, as it was generally agreed. This is pure exposition, so I was probably a bit naughty, but I thought it was kind of cool to sneak it in. Nico doesn't mention it here, but the body when it lay in state awaiting burial was always placed with its feet towards the door, which was to prevent the dead man's psyche from wandering off.
The psyche was the closest thing the Greeks had to a ghost. It was possible, if burial hadn't been performed properly, for the psyche to hang around. But this was extremely rare because the Greeks had enormous respect for the dead, even of their enemies. (Which is what made Achilles' mistreatment of Hector's body so very shocking.)
A lot of people think the coin placed in the mouth was to pay to get over the river Styx. Nope. Hades had many rivers, and the first of them was the Acheron. That's the one Charon the Ferryman carries you over. The Styx was the end of the line; when you crossed it, you'd reached Hades.
The coin placed in the mouth was an obol, not a drachma. Six obols make a single drachma, which was the average daily wage in Athens. So an obol represents about 2 hours of work and is probably what a for-real ferry crossing might have cost at the time. It also meant anyone could afford to get their loved ones to their eternal home.
There was no concept in Greek religion of being judged after death. Good or bad, you ended up at the same place.