I feel like it should stay the same. All of Godzilla's history is based around his origin, and the fact that he is a result of man abusing nature (or in GMK's case, forgetting about those who die and suffer during war). Godzilla 2000 poses a question that I like. "Science has created Godzilla, and ever since then we have tried destroying it. So why does Godzilla keep protecting us?" I feel like if we change the history of Godzilla, he won't be the same thing he used to be. If he had any other origin story, it'd be another monster. Godzilla is the way he is, not only his looks and powers, but by his backstory. "Godzilla is a monster created as a result of man using nuclear weapons, power to destroy anything. Godzilla is nature's wrath against man."
If Godzilla had any different backstory, it wouldn't be the same. Like the new one, he was just a dinosaur asleep, woken up by a submarine. Not quite the same, and as a result he is now a hero. Not the anti-hero or a kind-of hero like he was.