This … is a superhero campaign in which the PCs aren't the superheroes. No, they're the people trying to survive a world in which a super battle might destroy their coffee shop, their car might get stepped on by a rampaging tyrannosaur, and their pizza delivery might come by way of super-speedster.
So what kind of life do normals live in a world where supers take care of everything? Well, it turns out that the 'super crimes' get all the attention and glory, which means that ordinary criminals can do quite well as long as someone else has a flashier gimmick. And ordinary cops get no respect or attention, because they can't deal with the freak with six arms shooting up the town.
Honestly, I'd like to run a cops campaign in such a world, but I'm open to other directions players might want to take it – the only restriction is that none of the PCs have superpowers, nor are they close friends with anyone that is.