I've always wondered whether they SHOULD. From my point of view, I don't think they should. Especially Superman. I feel that Superman should always find a better solution. That's what makes him Superman: he always does the right thing. Oddly enough, I'm okay with some Marvel characters taking a life. Captain America fought in WW2, so I don't see why he shouldn't. It would pain him to do it at any time and he'd avoid it as much as humanly possible, though.
I think this is less than a case of "should the character ever kill" and more of a "should the writers be telling a story where the character kills". It's established, and agreed on by most fans, that there are circumstances that would cause Superman to kill, and sometimes those stories work, but for the most part it's much better to tell the story where Superman finds a way to not kill. It's better to tell a story where Batman doesn't have to resort to a gun.
I think comic book movies fall into a trap where they feel like they have to push their characters to the breaking point. Big stuff needs to happen on the big screen, and I think it's lazy writing to go straight to pushing a character to the limits of their morality, especially if that morality isn't well established onscreen.
I've used this analogy before, but here goes: Nancy Drew stories are about her solving mysteries. Is there a mystery that Nancy Drew can't solve? Well, obviously, she's only human, there are limits to her abilities. Writers could come up with a mystery that she couldn't solve. They could come up with a mystery where she gets it wrong and people suffer because of that. It could even be an interesting story. But it would be
absolute shit as an origin story movie on the big screen. Even if the writing were excellent, even if they somehow managed to portray her as competent in a "show, don't tell" manner, it would still be a crappy Nancy Drew story (at least as far as a stand-alone movie, or the first of a series of movies). A good first Nancy Drew movie would have her solving the case, because that's what the character is about. It could be more nuanced than just getting it right and that's it, but the character is about solving mysteries, and if the mystery goes unsolved then what the hell is the point of making it a Nancy Drew movie?
You don't make an Aquaman movie where there are no underwater scenes.
You don't make a (first) Star Trek movie where they stay on Earth.
And you shouldn't make a Superman movie where he kills.
Yes, you could tell a really good Aquaman story set in the desert, but don't make that your movie. Yes, Star Trek 4 was basically set on Earth, but it would not have been a good choice for a first movie. So, while you can tell a good story where Superman kills, it shouldn't be your choice for the first movie in a series, and it shouldn't be done too often or too lightly.