See, that right there justifies the existence of an anti-supers movement - if these destructive individuals don't have control over their actions then you have a pack of mad dogs with antipersonnel missiles in their back pockets. At that point you've depersonalized the Supers into a thing rather than people, and that's a really problematic trope in fiction. (see Starship Troopers, any wartime propaganda, most human-vs-machine conflicts...)
I think it makes for a better worldbuilding concept if it's not something inherent in the Supers that guides their actions, but rather that these are normal human psyches suddenly given capabilities way outside of the box. There are a lot of interesting analogues this could bring forward: if an individual possesses the ability to kill hundreds of humans with superpowers, what safeguards need to be put in place for the common good? This could parallel real-world situations such as "if an individual has enough wealth to destabilize and disenfranchise hundreds or thousands of humans by their decisions, what safeguards need to be put in place for the common good?"
You'd find arguments like "Can there exist a truly good Super?" in the way that we have real-world arguments like "Can ethical billionaires exist?" - think about how certain writers emphasize Superman's alien nature: yes, he uses his abilities for altruistic purposes and the protection of humanity, but what if he didn't? - this is the justification for Lex Luthor's actions, most of the time.
So I imagine you'd see a lot of Supers working either in secret or obscuring their identities simply because there would be an inherent fear and distrust. Nietzsche wrote about a similar concept in Thus Spoke Zarathustra from the spiritual and intellectual perspective, the "Overman" who would use his enlightened mind for the common good but still be distrusted by the Common Man for his abilities; contrasted with the "Last Man" who would use that enlightenment for pure self-interest and justify the Common Man's antipathy.
JMS's Rising Stars comic really leaned into this - superheroes are cool, until they aren't. They're trustworthy, until they're not. If you're a Super who can blend in with the population, why would you want to submit yourself to that distrust and antipathy unless you really don't consider yourself part of common humanity? Down that way lies your Homelander psychosis - "I'm a god, everyone else is just insects".
This is basic comic book psychology, the idea that superheroic identities exist to portray supers as mythical, as more concept than man, because people can believe concepts and caricatures to be free of human foibles and weaknesses. If Stupendous Man flies in to save an airliner from crashing, people accept it - that's what Stupendous Man does. If Doug Gruberman flies in and saves an airliner from crashing, people want to know what this guy's motives are. Does he have stock in the airline? What if other airlines crash more often because they don't have a superhuman looking out for them?
I think humanity would almost have to create the fiction of superheroes and heroic alter egos to insulate themselves from the existential dread that the presence of supers would bring with it.
