I think that you need both. The reason why a story about Clark Kent facing ordinary problems works is because of the contrast: “Here’s a guy who can bench-press an aircraft carrier, and yet he still has a hard time asking out the girl he likes”, or whatever. And yes, to make that work, you do have to sometimes show him bench-pressing an aircraft carrier. But it should be the background, not the foreground.
And this does vary by character. Fight scenes involving Superman are only superficially interesting, because you already know how he’s going to win: He’s stronger, tougher, faster, mightier than his opponents. If he needs to take a punch, he can. If he needs to punch someone else harder than they can take, he can do that too.
But if you look at, say, Spider-Man, while he’s also superpowered, he has definite limits. He sometimes gets into fights that are beyond his weight class, such that if he tried to fight like Superman, he’d get splattered. And so he needs to figure out how to leverage the powers he has, and use them in clever ways, and that’s interesting in its own way.