I never said "Balance means killing Superman." I said Godzilla would seek him out as a disruption of balance. Natural balance dictates that all life that originates on Earth (whether by evolution or by design, you pick), grows and adapts to suit itself, and creates a sort of harmony. Yes, things die and animals kill and eat each other, many events that some humans look upon as cruel and unfair (like a wounded deer getting no help). That is the natural balance. When a scientist on a dying planet sends his infant here in a rocket, and that infant absorbs our sunlight and uses it to become far stronger and more destructive than any natural human or animal, that is a blatant unbalancing of nature. Superman's anatomy does not behave in any way like that of any living thing on Earth, and he uses it to defeat or destroy threats to humanity; which is using alien resources to solve native problems. Think about what happened when rabbits, which have no natural predator, were brought to Australia.
Rabbits in Australia - Wikipedia
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1. Nature has no inherent Balance to it.
2. There is nothing unnatural about Superman. He is a naturally born individual with abilities derived from his natural bodily functions.
3. Even if there were such a balance, by framing it as being in opposition to Superman's traditional activities, that Balance then becomes a villainous force that must be resisted, not acquiesced to.
The reason this rivalry sounds so good to me, now, is because they could fight each other again and again until they come to an understanding. One of them wouldn't just roflstomp the other and move on. This balance of power means that Godzilla could adapt to see Superman as a net benefit in the face of forces like Darkseid or Mongul or Sun Eaters. Superman could learn to keep a cease-fire with Godzilla, where the King of Monsters stays out of human cities unless needed to fight Kaiju, and Superman does his best to keep a handle on the destructive potential of his powers.
Except you're set up precludes them from ever coming to an understanding, because Superman is never going to stop using his powers to help people.
What big events? Do you have some ideas for a timeline or something, or are you just referring to the splurge of Superhero Movies post Avengers?
Maybe Godzilla is drawn to Metropolis because Lex Luthor is doing some weird radiation experiments (kryptonite or otherwise) that damages nature/the ocean. Then when Superman sees a giant lizard attack his hometown, he goes to fight, which is the start of a series of massive battles over months or maybe years.
I don't think you need a special reason for Godzilla and Superman to fight. Godzilla is a monster lashing out at humanity, and Superman is a Hero who wants to stop him. That's how they meet, the deeper connections develop from there.
Here's how I see it going:
There first battle happens when Godzilla attacks a city, motivated by his still burning anger against humanity. Superman intervenes in order to preserve lives, and the fight ensues. Godzilla's sheer might forces Superman to progress into lethal force, culminating in him beating back Godzilla's atomic breath a la the ray gun from the Fleischer shorts. Godzilla is seemingly killed, but so is Superman. From there the events of Reign of the Supermen occur, and Superman eventually returns, having realized that he was wrong to use Lethal Force, and that he must hold to his ideals even in extreme situations. Knowing that he will eventually need to face Godzilla again, Superman begins work on a device that will stabilize Godzilla's body, soothing his pain.
After a couple years, he manages to do so. During this time, Godzilla has begun mellowing out due to influence from outside kaiju like Mothra and Anguirus, bu when he sees Superman, he attacks in a rage. They fight for a bit, until Superman manages to activate his device. With the chronic Pain he'd been feeling for his entire life gone, Godzilla calms down and the fight ends peaceably. After this encounters, Godzilla adopts his son, and begins transitioning into a more heroic figure.
Years down the line, Godzilla as fully transitioned into a guardian a la the Legendary films, and has a final battle with Superman, after the ladder his been dosed with Red Kryptonite. The battle is intense, but Godzilla manages to Keep Superman busy long enough for the Red K to wear off