The problem with most of those Superman scenes is that nearly all of them include moments of Clark looking legitimately troubled by the counterpart each character makes, and not in just a "I'm thinking of how to respond to this way."
Superman shouldn't be troubled by humanity's cynicism, violence, hopelessness, and xenophobia? He should smile and appear encouraged in the face of hatred and despair? What? That's not how Diana reacted. It's okay to be angry or sad about things. Just because you are doesn't mean you're hopeless or apathetic.
Heck, even the saving montage was done to music that signaled dread, and each time Superman saved someone it was accompanied with shots of him looking into the distance as if he was questioning what he was doing.
The montage was done to mournful music because of how humanity was responding to Superman. The musical queue is literally a twist on the more hopeful queue from MoS, and it is signalling humanity's division over Superman. You are making something that is about US into something about HIM. That music reflects the darkness in humanity rather than the darkness in Superman. It is our existential crisis. The narration of the montage is quite literally the opposite of Superman questioning himself: it is humanity questioning him.
All of this then capped by the scenes where he says the House of El crest means nothing, and that no one stays good in this world, does not give me the impression that Superman was truly someone who believed in hope and inspiration, it gave me the sense that he was striving to, but hadn't convinced himself yet.
"You are strong because you are imperfect. You are wise because you have doubts." -- Clemmie to Winston Churchill in The Darkest Hour
Superman is simply stating the facts in front of him. The El crest means hope, but the Capitol bombing and its antecedents suggest Superman is not the beacon of hope Clark hoped he would be. He is remarking on the fact that humanity is struggling to find hope in Superman because of its own existential crisis. We, like Lex, are having a hard time confronting our own powerlessness in the face of power: If God is all powerful, can he be all good? And if he's all good, how can he be all powerful? He's not convinced humanity is ready for Superman, and due to the tragic events at the hearing, he's trying to figure out what is the best way to serve humanity without upending everything.
The most hopeful people I know are not immune to indecision and doubt. The hopeful person is someone who confronts the darkness and ultimately decides to continue to seek the light. Hope is revealed through action and attitude. Superman, in BvS, reveals himself as hopeful by ceaselessly searching for the light in the darkness. His trip to the snowy mountain to find his way through the nightmares and his olive branch to Batman before the fight show us hope in action.
But I wouldn't say he's presented as a cynic, I would say he's presented more as lost. As someone desperately searching for a belief system that he hopes to be true.
"All that is gold does not glitter,
Not all those who wander are lost"
-- The Lord of the Rings
I wouldn't describe him as lost at all. A person who is confronted with difficult questions and searches for answers to those questions isn't lost. A person who is confronted with difficult decisions and makes those decisions with deliberation and care is not lost. Introspection and care is what is necessary to navigate complex issues that could have immense existential consequences. Clark Kent is a journalist; the woman he loves is a journalist. Good journalists know that the best stories are those that come from having an open mind, responding to the facts, and finding the truths that shine through the shadows.
Superman in BvS is someone begins with confidence. He's had an 18 month love affair with the world: humanity has built statues to him and the media can't stop themselves from writing puff pieces about him. Clark doesn't care what people are saying about Nairomi. Senator Finch launches a committee to investigate Superman. He's a little taken aback, but it's still not something that is preoccupying. Then, Keefe calls him a "False God," which affects Clark, as it should. It would be heartless not to feel something in the face of Keefe's pain.
At this point, Clark has completely shifted any interest or concern about Superman toward his investigation of the Batman. For him, the choices one makes says a lot about what one values, so Clark chooses to invest his energy into helping the scared and forgotten people of Gotham who are terrified of a Batman who is hunting. He's so focused on this story that nothing else seems to matter. When Bruce hurls anti-Superman invective at Clark at the gala, Clark remarks that the rest of the world isn't as cynical. Clark leaves the gala to save a little girl from a fire, which begins a montage in which it becomes clear that the rest of the world is wrestling with his presence in their lives without giving in fully to the cynicism of Batman's point of view. Nevertheless, Superman does not doubt, retreat, lash out, or give up. Senator Finch invites Superman to testify and to confront his accusers, and he does. If good is a conversation, Superman puts his faith in the people and what is good by joining the conversation.
The only time Superman wavers in his conviction is after the Capitol bombing. Because Superman is the reason behind the tragedy. Whether it is Keefe, Luthor, or anyone else is irrelevant. Someone decided to bomb the hearing to make a statement about Superman. How can something so awful not be cause for reflection and doubt? But the true test of hero is what he does in the face of doubt and darkness. Clark decides to reevaluate his role, and he concludes that Superman should stay. Later, when confronted with a choice that could destroy him, he decided to take a leap of faith anyway. Even if no one stays good in this world, Superman was still going to try to see if he and Bruce could defy the odds, and they did.
And for the no kill thing...come on man, in the Byrne comics what I'm asking to happen with Zod is what actually took place. Superman went on a long self imposed banishment and did some major soul searching after killing Zod in the comics, returning with newly imposed moral codes. Ones he couldn't always follow, but it's certainly present in the comics. And again, at no point am I claiming he should NEVER kill. What I'm saying is that if you make it something he decides he's morally opposed to, and then challenge that, it's makes for dramatic weight. Or if you use a moment where he has killed to then further build his character, it adds dramatic weight.
And I'm saying that it isn't something Superman decides he is morally opposed to. So if he didn't make that decision, then why should the narrative challenge it? Superman's "no kill code" is often presented as a flaw in the comics. Most notably during the "Sacrifice" arc featuring the death of Max Lord at the hands of Wonder Woman. Superman was a self-righteous hypocrite. The soul searching in the comics following the death of Zod makes sense since that Superman killed Zod in cold blood. He acted as judge, jury, and executioner. He was morally wrong.
The fact that the Zod death was used for neither of these things, and the fact that we didn't have an established moral view of what Clark thought of using lethal force before or after Zod's death neutered that moment of further dramatic potential.
The drama is in the choice itself. The drama is the trolley problem. Superman killed Zod because he chose one life over the lives of a family, over the fate of all humanity. The sort of drama you're referring to only happens if the only way forward for the narrative that you can see is one that shapes or changes Superman. But what if the purpose of Zod's death is already achieved through the death itself. Killing Zod establishes Superman is willing to kill if there is no other way to preserve innocent life. There's nowhere else to go from there. It is what it is. It's only an unsatisfying answer if one is uncomfortable with the answer.
And yes, Alfred tells Bruce he thinks he's acting like a man in a fevered rage. In a conversation. Does that strike you like the actions of someone who actually thinks that? He then continues to further debate with him. We don't see Alfred try to take any action, we don't see Alfred acting as if this is extremely unusual for Bruce. We can tell he disagrees with it, but it's presented as if Alfred views this as extreme and he's trying to debate him out of it, not as if he honestly believes Bruce has gone off the deep end. And since we have no previous experience with this Batman to compare if this is unusual behavior for him, Alfred is our only measure for what is normal for this Bruce Wayne. And at the end of the day, he merely debates him and then goes along with his plan. That doesn't strike me as "this is clearly intended to be a Bruce Wayne who has gone far beyond what Alfred has ever experienced." If it was meant to be that, it was done very poorly.
As a counselor myself, I can tell you that the film nailed the PTSD narrative for Bruce. Not only did it take care to showcase Bruce exhibiting all of the diagnostic criteria for the disorder, but it also showed a realistic response from a loved one. Alfred is stuck between concern for Bruce and feeling like he needs to stay close to him, reason with him. The film clearly establishes that Alfred is concerned about Bruce. At no point does Alfred respond to Bruce as though he is acting rationally.
So while I think you're interpretation of what motivates Bruce and the PTSD angle is an interesting one, I ultimately don't think the film supports that. Which is unfortunate, because I actually think it's a very fascinating take. But again, you need to show us how this experience with Superman is different for Bruce personally, and WHY.
It is not interesting. It is the truth. I know people with PTSD, and I've studied PTSD. Every single element of Bruce's characterization in BvS supports a PTSD reading. He has all of the symptoms (e.g. reliving the event, nightmares, flashbacks, triggers, avoiding seeking help, negative changes in beliefs, negative changes in behavior, difficulty sleeping, isolating oneself, seeing the world as dangerous, seeing everyone as untrustworthy) and the trigger for his recovery is quite similar to the effects of a common therapy for PTSD (e.g. exposure therapy, reliving the traumatic event, face your fears).