Well, there are some actual Lex Luthors out there. Some of them very high profile and currently trying to stay out of jail. Not all our Luthors have that Luthor level of intelligence, thank the gods, but they're just as rich, influential, and evil.
But I get what you mean. If a Superman shows up and saves the world from a Brainiac, that changes the equation. If the human race avoids total extinction from an outside force, that would earn a Superman some points. Not enough to make him a wholly accepted and trusted entity, not by any metric, but it'd help. But if it's just a Superman, with none of the Super-sized threats?
Chaos, riots, and the impotent rage of the powerful discovering they're just puny humans after all.
Personally I'd love to see it. I'd bring hot dogs and marshmallows to cook over the fire as the world burned.
But I don't think even a narrowly defeated Brainiac would earn a Superman full acceptance. No matter what he did, people would find fault with it. Politicians would try to force it into their dogma, either for or against depending on how they wanted to spin the PR, nations would ramp up defense spending to the detriment of the people (who's ready for another economic recession because we're buying too many bombs and not enough school supplies!?) and all that before we consider the depth of this Superman's actions. If the guy decides to deal with climate change or some other big issue, nobody in power would support him and they'd try to ensure none of their supporters/citizens did either.
Just look at how some folks respond to young climate activist Greta Thunberg; that's just a kid being mouthy about a topic any moron can see is a legitimate concern, and there's plenty of folks who want her in jail, if not accused of treason. Imagine if she had the power to actually do something about it? She'd be labeled public enemy #1; too many rich people would be afraid of losing money. A Superman would face the same problem; any issue they try to take on would end up taking money from some rich scumbag, who would in turn try to paint this Superman as a dire threat to civilization.
Hell, people can't agree on which side the toilet paper is supposed to roll off from (against the wall or front-facing? The debate continues!) there's no way in hell the world would happily get in line and universally support an alien trying to force their opinions on us.
And I disagree with manwhohaseverything on the religious aspect. There'd be some people who would see a Superman and think it was a divine intervention. But there's just as many who would call it the work of whatever devil they believe in. And it were known that this Superman was an alien? That throws basically every religious teaching on earth out the window. No faith would happily accept the guy who disproves them.