Why can't Superman kill? You can't just say "because he's SUPERMAN." Characters need a motivation. You can't have Superman refuse to kill nearly unstoppable villains who aim to murder everyone in their path, and justify it by saying, "well Superman doesn't kill." He needs a reason to be willing to put the world at risk by keeping these dangerous villains alive. And you know what's a good reason? Having Superman brutally murder the first villain he faces because there is literally no other way to stop him.
You need to start somewhere. If you want Superman to be against killing villains, then you need to give him a specific reason for this mindset. That is where rules come from. Rules are made only AFTER something bad happens. There needs to be some kind of precedent. Superman is not born with a rule against killing. He has that rule because he HAS killed, and he knows it's something he needs to prevent in the future.
Also, let's do a comparison:
Superman 2 - Zod, Ursa, and Non are all DEPOWERED. They no longer pose a threat. Superman could easily fly them to a prison, or put them in a cage in the fortress. But no, he tosses Zod off a cliff in the fortress. Then Non falls. Then Lois pushes Ursa off the cliff too. Obviously these three depowered villains fall to their deaths. No one was in danger anymore. They were powerless. Yet Superman goes out of his way to kill them. How does Superman respond? He smiles, and Lois smiles, and they hug.
Man of Steel - There is no way to stop Zod. Zod stops giving a ****, has a deathwish, and now exists only to kill humans until Superman kills him. Zod is about to kill three people, Superman's only options are A) let him kill the humans, B) somehow get Zod away from them and then follow Zod around the planet stopping him from killing people for the next billion years, or C) kill Zod to save millions of lives and end the destruction. Don't even try to tell me the first two options are viable. When Superman does kill Zod, his reacts very strongly. He is devastated. The subject is treated seriously, and not seen as a time for celebration.
Say what you will about the neck snap, but at least this Superman didn't kill a powerless Zod for no reason and then smile about it.