It's an amusing enough hypothetical, but I'm gonna take it in a different direction. first off, Superman fucking sucks. I never liked the "I'm GOD" version of Superman where he could shrug off atomic bombs and only need to comb his hair afterwards. Originally he was only a "super-man", superhuman in all aspects but otherwise a normal guy, just better. Bigger, stronger, faster than a speeding bullet and all that. He couldn't even fly, if you'll remember...that's why the famous radio drama line talks about him leaping tall buildings, but I digress.
The current version of Superman is that he can fly. He also can melt people with his eyes, just by looking at them. He can hear things from miles away. He can see through walls. He can freeze people in blocks of solid ice by simply breathing on them. He can punch through walls, pick cars up over his head, and bullets bounce off of him without leaving a mark. All of this is so trivial it's not even up for debate. The only argument comes when trying to define the upper limits of his power, such as whether he can fly through a star, how much damage does he take when he gets hit with nuclear weaponry, and other such absurdity.
See, this is why I hate the character. Most characters have weaknesses. They have flaws. Batman, for example, is just a normal guy who could take a bullet from any random thug on any random night. He gets beat up regularly because some of his foes have legitimate superpowers while he does not. There's actual risk involved because they can beat him, and that's why Batman has to be the best to always come out on top. Superman's just some lame video game character on cheat mode.
All that being said, I've always wondered why Superman ever even had to fight crime. Outside of the very few other god-like beings on his level, no one else can match him and as soon as the Ultimate Boyscout shows up you'd think everyone would just drop their guns and give up on the spot without a fight. I mean, it's fucking Superman - what the fuck is a common street hood like Rocko or Twitch gonna do against him? His very presence means it's automatically "Game Over".
Taking it one step further...the people of Metropolis all know that there is a superbeing living amongst them that basically is a living demigod. He isn't all-knowing, but he has super senses and a super sense of justice. Most people don't break the law out of fear of getting caught, and that's by mundane policemen. With a veritable deity amongst you, would anyone ever dare to step out of line? Even if he's not within sight, who's to say that he's not ten miles away and would hear you breaking into a car or hear a victim crying out for help? Again, his very presence...or the mere threat of his presence...is kind of a game-changer.
Maybe he's just in his boxers, watching the evening news at home while munching on some crackers? All he'll really need are the names and faces from the evening news of who's been naughty that evening, and within about an hour he could round them all up, kick all their asses, and be back in time for the nightly sitcoms. It's not quite as efficient as having a Death Note as he'd actually have to get out of his chair, but there's nothing keeping Clark Kent from becoming a costumed Kira. Obey the law or else the god living among you will come and punish you in the name of justice. There is no escape because he is all powerful.
A newsroom is the next best thing to having connections with a police task force. It wouldn't take long at all before everyone in Metropolis straightened up and started acting right because they knew that Clark Kira was always there enforcing the law and fighting for justice. To be honest, I'm kind of surprised that it isn't that way already.