[archduck]

The problem of evil arises when all three of the following are true: 1) the deity is omniscient; 2) the deity is omnipotent; 3) the deity is omnibenevolent. If God knows all, can do all, and is all good, why does evil exist?

If any of those conditions is not true, the existence of evil is a lot easier to explain: - omniscient and omnipotent, but not 100% morally good - omniscient and omnibenevolent, but has limited power to change the bad things - omipotent and omnibenevolent, but isn't aware that evil exists

It's only when someone's making the claim that all three are true at the same time that it starts to get interesting, because some of the arguments get really hokey.

So to address your observation that there's an implicit assumption that one god exists, I would wager a guess that it's because polytheistic religions don't wade into the waters of theodicy much (if at all), as they don't deal in "omni-'s" to the same extent as monotheistic religions tend to.