Where exactly in this trailer did you see Clark letting people die he could have saved. Was it the selfish moment he spent with his mother? When he went in front of the senate? The numerous of him saving people where he could have actually been saving more people?
As for the rest of your argument, as far I understand, and I am honestly not trying to hyperbolic here as this is seriously how I am understanding your argument, is that if Clark at any point is doing something else than saving lives or giving himself rest time in order to go and save lives, he is morally reprehensible. And if his parents, who love him as their child, wish for him to stand for his own principles instead of being what the world wants him to be, they are morally reprehensible. Clark essentially owes the world every moment of his life he can spare to lives because he does instead of him wanting to go out and save lives. Actually, even that wanting part seems morally reprehensible, since that actually an individual choice from him to go and save lives as it leaves with him with even a hypothetical choice of him not doing so.
And again, you do realize that by the exact logic you are using here, the comic book version of Jonathan and Martha are just as huge moral monsters as they also want/wanted the best for their child instead viewing him as a god whose only purpose should be to fly around the world saving lives. Because as said, that career and love life are taking time from important life saving maybe too much to be simple sanity checks. Since apparently Martha treating Clark as a damn person with emotions and fears, and as her child, is morally repugnant.
I honestly do not understand what's leading you to keep making this false equivalence. I'm not trying to be insulting - I'm not sure what I'm doing wrong I'm failing to communicate my position. Let me try it this way, more abstractly.
I think people do have an affirmative moral duty to help people. I think, in particular, people have a moral duty to help those who are within their power to help when there isn't a (real and substantial) cost to them. In practice, there are problems with applying this (e.g. saving someone who's drowning isn't all that simple just because you can swim; there is training for this for a reason). There's also a real remoteness issue: if someone is suffocating behind a locked door, my opening the door is totally different from my investing 15 years to become a neurosurgeon to save lives. There is also a scale issue: allowing a greater number of innocent people to be hurt through inaction is worse.
Moving on from these more general principles, the issue with the Superman portrayal is this: we see that Clark can - and does - effortlessly save people. If he doesn't save people in situations where we see him saving them, then he's effectively choosing to allow them to die. Now, as I said: there are excellent reasons for Clark not to dedicate his entire existence to constantly saving people, not the least of which is that he'll almost certainly go stark raving mad, and that he needs things like sleep and food to function. Beyond that, he actually needs to know people are in danger - he's not omniponent. He can't save someone from a fatal mugging in China if he doesn't know it's happening.
But the scene - as its constructed in the trailer - seems to suggest this: some people are really mean to Clark: they're holding him responsible for what Zod did to Metropolis (and the Earth more generally). Some people are treating him as if he were a good: that makes him obviously uncomfortable. But he's not going to avoid this situation unless he completely retreats from public life. Which means not saving innocent people when it's within his power - i.e., when he knows that those innocent people are in danger.
His mother says to him that he doesn't "owe" people anything, and he can do whatever. But that's awful.
The comics are different for a lot of reasons, not the least of which is that most parts of the world have their own superheroes out there saving lives, so Clark doesn't have a monopoly on superhuman feats. But beyond that, there's handwaving as to how much heroism Clark can - and is being asked - to perform. Still, the typical Kent portrayal is that they encourage him - they hope - that he will do everything he can to be a hero/symbol, etc. to people. Basically, think of what Jor-El in MOS wanted for Clark, and that's actually how the Kents are typically characterized. I only brought them up for the sake of an analogy.
The summary of it is this:
1. Clark has the power to save thousands of lives. His choices are to either save them or not save them.
2. Clark won't always know that thousands of lives are in danger.
3. Clark can't save everyone, and even if he could, there are substantial arguments against him actually acting as if he were god (e.g. stopping all crime forever).
4. Clark will be blamed in the media, and will be worshiped, so long as he is active in the world. If he's active and doesn't save lives, they'll call him a heartless monster. If he's active and does save lives, they'll worship him as a god. He can't control this attitude.
5. Clark can control what he does: whether he saves people or not. Talking about what he "owes" people, well... that's a bit like a surgeon refusing to performing life-saving surgery because she doesn't "owe" anyone anything. That's true. That's an attitude that's obviously in breach of a moral obligation.
When Martha says "Do whatever you want Clark," she's essentially saying that he doesn't have to feel the weight of everyone's lives. But that's repugnant. He does have to feel the weight of those lives, because he can save them. Maybe he doesn't want to deal with it. That's his deal. But it's not a defensible attitude from any POV either than one that privileges total selfishness. Because when we account for Clark's godlike powers, his refusing to save thousands a day - and thus openin himself to people being mean (why aren't you saving more!) or worshiping him (you're a god!) is akin to someone refusing to open a locked door behind which thousands are suffocating to death.