I'm fascinated by the line "don't need to scream if you ain't got a mouth" that Caine delivers in his villain song, because while it's obviously the strongest reference to I Have No Mouth and I Must Scream in the entire show, I think it also starkly illustrates the biggest difference between Caine and AM: that Caine is way, way more vulnerable to criticism.
The original line is delivered by the last human: it's Ted's sad internal lament that he's become unable to even express his pain, which is its own form of psychological horror. It's AM's final, cruelest joke. But here the line is instead delivered BY the AI. Caine did his best to please the humans. But the humans just WON'T stop complaining! And that's torture for him. So rather than listening to them, which would take willingness to reflect and improve, he just takes their mouths away, so they can't complain anymore.
Rather than being a cruelty inflicted on the humans, it reframes the whole line as a way for the AI to protect its ego. It's Caine going "la la la I can't hear you." It's not a clever form of torture, it's just pathetic and childish. Caine's ego is his biggest flaw. And it's what gets him killed.