On Guilt
Generally, the way morality and condemnation works is that it relies upon agency or free will. A morally culpable action is required for a person reasonably to be saddled with blame (disregard the unreasonable for now, there are less logical restrictions on bigotry).
Additionally, for unintentional harms we segregate cause-in-fact from the concept of proximate cause... that the thing that causes the harm needs to be sufficiently tied in time, place, and expectation that your actions would likely lead to the expected harm. If you stick your feet out carelessly while reading the newspaper on a bus bench and trip someone, you could be said to be the proximate cause of the fall. However, if your postman briefly blocks your driveway, such that you catch six red lights, and get hit by a drunk driver... your postman would be the cause-in-fact, you could say, "If he weren't there blocking my driveway, none of this would have happened!" BUT reasonable individuals would recognize he isn't the proximate cause and would not bear moral culpability for your car accident. Additionally, since the drunk's intentions resulted in the accident, he would be the superseding cause for the accident.
Humanity will have some degree of answers as to how Superman arrived on Earth and a big part of that disclosure is that he came as an infant, one who had no choice in the matter. Having no freewill or agency in his presence, Kal-El would not be morally culpable for his presence. There might be some argument of negligence with respect to Clark going onto the scout ship, however, again there was no intention to summon Zod, no knowledge Zod existed (Jor-El explained their world was dead), no expectation a beacon would go off, and irrespective of Clark's actions the ship was about to be discovered and investigated anyways which could have possibly resulted in the beacon going off regardless (as an intervening cause). So Clark would not have been a proximate cause Zod's actions.
That, of course, is the largest point, that Zod had a degree of agency in his actions. When someone else acts with evil intent, it trumps any negligence on behalf of a prior individual (unless their negligence could be expected to bring about the evil- like leaving a door unlocked leading to theft)... Clark would have no reasonable expectation that simply existing would result in another individual attempting to exterminate humanity.
Thus, under traditional morality, most people judging him would not assign guilt. Irrational hatred, surely, could be directed at him, but what does that accomplish for Luthor if he fans that flame? Even if he can whip hatred into a fervor that results in Superman being criminalized (just for existing), exiled, and shunned... what does that accomplish if you can't enforce it?
---
Blame Game
Irrational hatred is a part of humanity but difficult to sustainably monetize or politicize. The KKK is an extremist joke today and can be discounted. In first-world democracies governed by law and order, tolerance (however troubled and with work to do) is mainstream. Creating a "rift" doesn't accomplish anything.
Blaming Kal-El because he was the target of Zod's focus is like blaming America for building the World Trade Center to be targeted... it's not something you can get traction on with anybody of significance or influence.
It works with the X-Men because the group includes individuals who have actively and proudly joined terrorist organizations to abuse their abilities (versus a single individual who has done nothing to justify moral condemnation). For the X-Men, there is sufficient precedent to make hay even if the enactments are overreaching AND you can effectively enforce those enactments because, by and large, mutants don't have the same scope of powers as Superman.
If, somehow, under crazy bigot congress theory, Lex gets legislation passed against Superman... what does that do? How are you going to stop him from saving hurricane survivors? Is that going to stop invitations from Canada and the rest of the free world pouring in invitations to Superman? (We've already seen this version of Superman is happy to reside in Canada....)
It's a short sighted strategy that doesn't get Lex anything. He'd need the entire global united against Superman and the collateral damage to Metropolis doesn't get you there, not when Superman saves the entire rest of the planet and not when the planet can't do anything against Superman to back up their enactments. This is a Superman willing to down a 12 million dollar drone just to make a point... imagine what he'll do if you try to pass lame duck legislation against him (from Luthor's perspective: whipping up hatred just to get to the point of passing ineffective law?).
Conversely, what does allying with Superman get you? Potentially substantially more... Lex gets to frame the first contact narrative, he gets to be seen as the foremost authority on extraterrestrials, he gets to be perceived as mainstream and thus congress with actual big movers and shakers (rather than sound like an extremist nut job), he gets into Superman's inner circle to learn his vulnerabilities, he potentially gets legitimate access to alien technology and testing of Superman's abilities (assuming consent), and he gets defense contracts if he justifies why the world needs more than Superman.
My concern is that hate-speech version of Luthor is going to turn him into J. Jonah Jameson... basically a joke. You can suspend your disbelief for a little bit and see Luthor getting some traction with hate, but at the end of the day, hurting someone's feelings isn't productive and doesn't accomplish anything for a business magnate.
---
Above posts came from another discussion massaged to fit, but to address your point more specifically, I think you're skipping the impact of a first contact. The Kryptonians give humanity cause to unite and prepare against future incursion. This time they were spared by utter happenstance- an allied alien raised amongst them fighting a militarily crippled and resource starved invading force... why should they anticipate being as lucky the next time out? Now that the paradigm is that technologically superior races exist beyond their own, why would humanity assume there are only two? (Again, this relates to disclosure... we don't know how much was conveyed to Kal-El and then to humanity... but in the prequel comic, the Kryptonians- at a minimum- were aware of the Thanagarians and humans.)
As this relates to Lex...
---
Q: How do you speak ill of an invulnerable alien who's faster than a speeding bullet, can see your secrets through walls, can hear your every whisper, can incinerate you with just a glance, or disappear you in orbit without anyone knowing?
A: Very carefully.
I don't think a smart Lex can exploit xenophobia the way people are assuming he can... at least, not in a credible fashion. The only way they can speak against Superman is if they earnestly believe he won't hurt them (and have valid support for that belief), undercutting their entire message. Killing Zod sets a precedent that lets the filmmakers refresh the Lex Luthor character so that he isn't tritely trying to discredit Superman in public like he traditionally has done in the last few decades of the comics.
If Lex is Machiavelli, you need a bruiser, so I think Metallo provides the primary conflict so that you don't build up corporate genius Lex just to turn him into a mustache twirler or powersuit psycho in one film.
I think you use Lex to play xenophobia more subtly rather than hate-mongering causation. Lex goes to the governments of the world with Superman at his side and says something like this:
"We are not alone. As we know from that fateful day, the universe contains other souls, some as noble as our Friend here. However, as we all experienced, for every one like our Friend there are many more who might be our enemy. Thank goodness for him, or we would have been crushed under the boot of an alien people with technology and capabilities far greater than our own... but not beyond us. While I'm grateful for my Friend, while we all are, Humanity just barely escaped extinction. The day may come when we have more visitors from the stars and if they come in peace, we will welcome them like our Friend... but if they come to conquer or kill we cannot let the entire burden rest with our Friend. I call upon the people of this planet to step forward into tomorrow. Together, nothing is beyond us." (Except not horribly cheesy and terribly written like that!

)
With funding and multi-national backing, Lex can advance the transhuman project of making an immortal, superior man of steel / man of tomorrow to serve as a future foot soldier against aliens... foreign or domestic. If Superman is on board, there might even be some "friendly" sparring before it all goes haywire.
The primary themes of the film would be trust and humanity.
---
I hope Lex endorses Superman because I envision a Lex that is brilliant and recognizes public hatred of Superman is not sustainable.
Even if Lex privately hates Superman and even if this is the perfect opportunity to harm Superman's reputation even further... what does that get him? How does that accomplish anything? He simply makes Superman feel bad and go to ground, making him even more elusive and harder to study or hurt. You have this invulnerable alien that you can't do anything about, so how is hurting Superman's feelings productive? Not to mention, you have to assume apart from the Black Zero event, Superman is doing good. Assuming Lex isn't stupid, he'll take Superman's earnestness at face value and recognize that Superman will continue to act in a way that's consistent with good. That means Lex can only sabotage Superman (which would be hell if Lex got caught), yell at Superman's do-gooding irrationally such that only fringe xenophobics listen, or he can be smart... he can keep his enemy closer. Legitimate businesses / political candidates / etc. can only go so far on hate and Lex would see that.
If Lex endorses Superman, assists in improving Superman's image, Superman is more inclined to be out there in the public, to disclose more about himself, and to subject himself to say insightful experimentation (if Lex poses as a friend wanting Superman to know his own limits so Superman can optimize helping people; you don't want to learn in the field what you can and can't lift, etc). Now Lex is gathering meaningful data and setting up Superman for a fall; he's getting inside Superman's circle of trust to learn how to hurt him physically and psychically... moreover, Lex maintains his legitimacy with the public because he isn't just ranting against an obvious, if dangerous, do-gooder.
As far as film tone, the other important thing about this is that it provides the treachery of man in a way that is not psychotic, wounded, or an easy bandwagon to jump on. If a quasi-legitimate billionaire suddenly dedicates his being to publicly destroying an invincible do-gooding alien, that's insane or woundedness... I want Lex to be authoritative, together, and brilliant. Regarding the bandwagon, there's no doubt that MoS has been controversial and has a number of detractors... you don't want a suave, charismatic, and logical villain actually convince the audience Superman is terrible before Superman's earned the audience's trust. Instead, you use that incredible actor to help earn the public and the film's audience's trust... then when Lex betrays Supes, the audience feels the betrayal too and Lex is the proper villain. You do not want Lex so sympathetic and logical that the audience is rooting for him to beat Supes!
Anyways, the idea needs more work but that's the kernel of it....
As a quick aside, even if Lex has person vendetta against Superman, I do not want that vendetta to be all consuming... I want Lex to be a fully formed character with separate ambitions and machinations. He didn't build his fortune all his life because he was shiftless and waiting for an alien to hate... he has his own goals. So this ties back to my underlying premise, if you come out as a hater, he's shortening his life as a legitimate businessman... instead, I'd like to see him play the long con and, at least publicly, endorse Superman.
Here's another way to look at it... traditionally, Lex knows he can badmouth Superman because he knows Superman will always hold back. How does this Lex know that? Superman's been in business for only a day or so and in that time he's been a part of leveling buildings and killing a super-villain... what kind of death wish would Lex have to have to want to make Superman his public enemy? If Lex were Superman, his critics would find themselves on one-way trips to the Sun. Lex can only speak out against Superman if Superman has already established himself as not-a-danger, in which case, Lex speaking out is ineffective (everything he spews just proves how much Superman will restrain himself). So I vote for a smarter Lex to sees this as a losing game and tries to support Superman publicly while planning against him in secret.