It'd only work if he signed on with the UN. He can't work for any one nation's government without eventually being co-opted into their military the same way Dr. Manhattan was, and he can't do anything without government support - someone has to trial and imprison the criminals he captures, and they'll never be convicted if he stays a vigilante. So he'd have to go to the UN.
Then they'd have a United Nations Superman Council, or whatever, and only the most powerful militaries get to sit on it, and they'll just veto each other all day long about where they're going to deploy Superman. Eventually he's going to be forced to hover above Bosnia or Uganda or Syria and watch people slaughter each other without the authorisation to go in, and he'll go insane - knowing that he has the power to stop it all, if only these stupid humans would let him.
Then he'd just go ahead and do it, and the UNSC would go berserk over their uncontrolled superweapon, and start fighting with each other over who benefits most from Superman's independent actions. If he's stopping warlords in Africa fighting each other, he's equally justified in stopping Russia from invading the Ukraine. If he doesn't stop the US from invading Afghanistan, Russia's gonna call foul and accuse Superman of America-favouritism. Eventually someone's going to throw a nuke. Even if Superman stops it, the UN would fall apart and the fragile illusion of international law will dissolve, because none of them can agree on what to do with Superman. Eventually, they'd only reunite if they all signed a Superman Non-Proliferation Treaty, barring Superman from doing anything in any country.
Then Superman would be illegal. He'd fly to the Moon and declare it his home, and from his Moon Base he will stare down at us, judging us with the power of a god but never intervening, even when he could stop great atrocities with minimal effort, because we ourselves exiled him. Over the centuries, he will fade into myth, and people will write books and tell stories about Superman, and the weak and the downtrodden will pray to Superman, and tell each other that Superman will return one day to deliver humanity from evil, and the faithful who lived their lives the way Superman wanted them to will be raised up to live with him on his Moon Base.
Eventually someone will ask why, if Superman truly has superpowers and is watching us all, does evil still exist? Surely he could stop evil with his heat vision or his super strength or his ability to fly backwards in time? And the masses will ignore the question, or spin rhetoric around it, and never stoop to answering it, because they know deep down that evil is something humans do to each other, and Superman cannot defeat evil without first defeating us.
