Models trained to do chain-of-thought do it without any specific prompting, and a specific system prompt that enables this behavior is often part of the setup,

Models trained to do chain-of-thought do it without any specific prompting, and a specific system prompt that enables this behavior is often part of the setup, because that is what the model was trained on. I don't see how that is "regurgitation", either, if it performs the reasoning steps first, and only then arrives at the answer.