As it stands now there's no way for an eliminated player to keep tabs on the game once they're gone. It shows the map as it was when they were eliminated, and that's it. If playing with normal fog, it doesn't show anything at all.

Is there a setting I missed or a mod that allows eliminated players to spectate an ongoing game?

You can change the fog setting to Light or No Fog, but this has serious implications on the gameplay.

There is also a spectator mod, which allows anyone to spectate the game (WARNING: Active players can use this mod to cheat!!)

UH how?

Spectating with an alt, they get extra information.

There's a really good, and really obvious reason that spectating players should not be able to see through the fog until the game is over. A spectator that can see through the fog can tell someone else what's behind the fog, giving that player an unfair advantage.

That's an argument against full spectating, not partial. There's no reason not to make public the knowledge that is known to all players in a game - eg in 1v1 the exact borders of the 2 players.

If the fog level is dense, light, or no fog, then spectators would be able to see territories' owners, and in the no fog case army numbers as well. In normal fog and dense fog, only the owners of the territories neighbouring the player can be seen, which for spectators is no territories at all. And of course, with complete fog you can only see territories you own, which is none when you're spectating. When the game's done for everyone, the entire map and all army numbers can be seen, as well as game history where you can see what went on behind the fog. Is game history not working? Is the game set to dense, light, or no fog, yet you can't see territory's owners? If the answer to both is no, then I don't see your problem.