Traditionally you don't start with one of those. You start with some sort of alien invasion or something that requires all the heroes to join forces. Most threats are local for these superheroes. Sure they can be called on to save the world, but usually they are just saving one city. Unless you are Superman, but that's because he's Superman.
And what you are looking for is the old Leagion of Doom from Superfriends. The 1970s campy version of the Justice League.
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