Is Superman brave?

I'd suggest that he does it because it's the right thing to do.

It's why lines like "Should I have let them die?" "Maybe." and "Be anything they need you to be. Or be none of it. You don't owe this world a thing, you never did." grate so much.

Apparently Snyder is something of an Ayn Rand objectivist. From what I've read, he finds Superman baffling because there's no obvious gain for him in going around saving people all the time. Which is why we get a Superman who doesn't seem to particularly want to do that in these movies.