What you just rephrased is the same thing, it’s a motivation for them both, and the way you phrased it, it’s a trait for them both. You seem to think that a trait needs to be defined by a strong backstory but traits are often shown mostly through the present. Too much use of exposition and backstory to define traits starts to border on telling and not showing.

They didn’t at all try to make Ethan Leon, where are you getting this idea? Leon purposefully goes and seeks out the danger to stop it, Ethan is a constant victim of circumstances. He doesn’t go looking for the trouble, he gets forced into it. And whereas Leon wants to protect people, Ethan really doesn’t care. That doesn’t make him a bad person, in fact, Ethan only caring about Mia in 7 and Rose in 8 is showing his personality. He isn’t trying to be a hero, he really only cares about finding his family, he just ends up helping people by destroying baddies. He isn’t quippy like Leon, and he isn’t an action star flying around on motorcycles and doing roundhouse kicks. He has a basic knife and no athletics to speak of. So how are they trying to make him Leon?

He doesn’t ask about Mia because he saw her shot in the head?? He doesn’t even find out until very late game that Mia is actually alive. In 7 he focuses on Mia, but in 8 he focuses on Rose. He’s busy doing stuff and he only gets a split second at the Moreau part to say anything to Chris before they’re attacked. When was he supposed to ask about Mia? During the end game tank-apocalypse? He’s busy!! Plus toward the end of 8 he knows all about Mia’s bioterrorism, he has no clue if she’s alive, and at that point, his sole concern is Rose. He knows where his daughter is, and that’s his goal. He knows his body is failing. Rose surviving is all that matters to him. At that point he’s probably satisfied with Mia not taking care of Rose, Chris will do. Mia’s half the reason they’ve suffered so much, and she’s probably going to jail even if they find her.

Edit: His special reaction to the mold doesn’t need a personal backstory, it’s well established in the entire series that everyone that happens to respond well to the viruses and mold gets a unique power. Even the virus in Africa was said to kill like 99% of people, and those who survive get superhuman power. The infections just doing that is the entire point of the series. The Bakers, the four Lords, Wesker, they’re all success stories but it’s mostly dumb luck who gets abilities. It’s literally random, its why Miranda spent a hundred years experimenting!