Warning: Spoilers for Suicide Squad: Blaze #1!
The Suicide Squad has assembled to take down a new villain, and on an ideological level, he could destroy Superman in a way Doomsday never could. This happens in the Black Label series Suicide Squad: Blaze. The series is written by Simon Spurrier and illustrated by Aaron Campbell.
The story starts off like any other Suicide Squad tale. A threat causes Amanda Waller to assemble a typical iteration of Task Force X comprised of members Peacemaker, Harley Quinn, King Shark, and Captain Boomerang. Where this story differs from most is that Waller also compiles five other non-super powered inmates and then gives them powers in a process called the Blaze. The downside is these powers significantly reduced the inmates’ lifespans. So essentially, the procedure is fatal.
The threat they are facing is a super-powered cannibal who is being driven by his base instincts to feed. When the new members of the Suicide Squad question their involvement, it's speculated that the threat this cannibal poses isn't just physical but also ideological. He is allowing the public to question the nature of superpowers and people are becoming suspicious of superheroes in the process. If left unchecked, this super-powered cannibal can tarnish the idea of a Superman so much that his reputation can take a hit that the hero might never recover from.
It's even mentioned in the book when a reporter asked Superman if he ever acts off of pure instinct. This is a psychological seed that's planted in the public's consciousness about whether super-powered people should even exist at all. It's an interesting dynamic because the only thing that allows Superman to be good is his public perception. Even with all his power, Superman doesn't have mind control. He can't make people like him. Of course, he can try to sway the public by force as is done in certain stories that have Superman become a fascist like dictator such as Injustice and Red Son. But at that point, Superman is no longer a hero, so it negates the argument.
This is the kind of moral rationalization that makes the Suicide Squad such an interesting team. On the one hand, they are being deployed to stop a cannibalistic superhuman simply for public relations purposes. Yet by doing so, they are also serving the public good by eliminating a threat that is killing people. They make mention during the book that they tag the creature with an isotope so that it is more easily tracked and then perhaps someone like Superman can come in and finish the job. Then again, this tricky situation of using convicts to accomplish this mission is a PR disaster in and of itself. At the end of the day though, members of the Suicide Squad don't necessarily care about the optics.