Elysium fields were restricted to the elites only.

That said, in the Odyssey, Achilles is very definitely not there. Likely as Homer didn't give any credence to the Elysium fields.

People assume that the Ancient Greek religion was like this orthodox thing, inline with say, the Catholic Church of the modern era. This is patently untrue. There were a lot of conflicting religions, theories, cults, etc. floating around. People would worship different gods in different places, believe in different afterlives (reincarnation was super popular, as it is in most Indo-European religions), etc. So it's hard to say exactly what a given ancient Greek would believe, as there wasn't a set religion per se. And people often *didn't* have problems with different religions the way we do. It was more like "OK, well, you have these gods that do X, we have a sort of equivalent god. Maybe they're the same, maybe not. Whatever. I guess we both pray and hope that we get rain or whatever."

That said, most of the gods were pretty batshit crazy, at least in existing myths (Homer, etc.)