Now, story-wise, the protagonists need to do something exciting, and the alternative plots are few and far between. (Exploring other planets or worlds not defined by planethood? Though even there they would probably fight.) But there are potentials for background characters, because unless the way you acquire powers is very focused, there is no way that everyone with powers wants to do it. (And even if everyone who gets them does, there's burnout.)
There's the obvious alternative of a superpowered character who holds down a mundane job in all seriousness. Wearing The Cape's Legal Eagle, whose flight power is nothing but a useful gimmick as he continues in his lawyer job, just tending to handle breakthrough-related law.
More interesting is the use of powers in non-fighting related jobs. In Wearing the Cape, Astra observed that most powers are useful only in fighting. This strikes me as improbable. Much depends on the powers, of course, but a person with a compelling voice could do the equivalent work of a hypnotist, and probably better. Flight would be useful for a janitor. Fire powers could be used for welding.
Kurt Busiek did an interesting bit in Astro City about superpowered characters who did other jobs. A telekinetic who did TV FX. A telepath who worked in Hollywood -- a director would hire him for his meetings with art people and instead of telling them what he wants, thinks of what he wants and has the telepath show them. One works in construction with superstrength. Of course, that story was about a villain who kidnapped a bunch of them to use as slaves. . . .
The one path to a "all fighting, all the time" universe would be if those who want to enslave them are too numerous and powerful for them to survive unless they train relentlessly against it, and then lack the time for a job requiring any other skills. More likely if they are deliberately made, and the makers want to use them to fight. Still, even there I suspect there would a transition to less fighting universe if the superpowered characters had any say in the matter, and any variation in their characters.
Interesting to ponder the variations.