I'm saying adding "think step by step" does not get you close to actual reasoning, it just produces marginally self-consistent linguistic reasoning.

Actual reasoning requires training on diverse data sources, as you noted, but also coached experimentation (supervised fine-tuning) not just adding "think step by step" instruction to a model trained on typical textual datasets. "Think step by step" came first and produced increased performance on a variety of tasks, but was overhyped in its approximation of reasoning.