> Not sure if modern models "think" only by outputting <thinking> blocks

That's pretty much it - a small refinement to "Chain of Thought" prompting, where you tell the model explicitly in the prompt to "Think step by step" or similar, so it writes out more steps before giving a final answer, potentially catching some errors. The "thinking" models are tuned to do that without being prompted to, and to output the "thinking" markers around it, so they can be hidden from the user.