Thinking about the Leverage team's relationships to Sterling and Chaos due to the random thought of "do they hate Sterling more than they hate Chaos?"
I do not have an answer to this question but I will now ramble about the different relationships the crew has with these two reoccurring villains.
When he are first introduced to Chaos he's tried to kill Sophie and honestly came fairly close.
"Long time now try to kill you" and all that. I feel like it's important to start with that.
He also successfully manipulated them into helping with a heist. They were later able to turn it around, and both those episodes ended with Chaos being arrested, but both probably weren't fun for the team and Sophie's near-death led to her leaving for Europe for several months to figure things out. It was good for her, but a lot for the team
And then of course there's the episode where they have to work with Chaos. It goes well, there's no double crossing or anything, but everyone including the new people hate the guy that whole episode.
And then there's Sterling. Who puts Nate in jail, forces them to blow up their first HQ, has gotten career boosts from them, once drugged Eliot, and successfully manipulated them, all this while trying to arrest them every time he appears in the show. There's a reason that this is Eliot's reaction to Tara saying "I hate this guy!"
He beats them multiple times and even though the team also succeeds, or comes back to succeed depending on the episode, that's really not fun.
On the other hand, this relationship is complicated by Nate and Sterling's relationship. They're not friends, but they respect each other, and Nate doesn't seem to hate him as much as the other's do. They also work with him 3 times, to rescue Maggie, to rescue daughter, and to clear Sophie's name. There's also the last episode, where he's pissed when he thinks Nate got the other's killed and lets them go when he figures things out.
So Nate likes Sterling way more than Chaos and the same can probably be said for Sophie because she likes messing with Sterling, while Parker Eliot and Hardison would probably be happy to punch and/or tase either one one.