I agree that LLMs don't really reason, but I'm starting to think that the name "Large Language Model" is just wrong enough to create this confusion. What they seem to be doing is modeling human reasoning as encoded in language, and then enable some mixing and matching of elements of that encoded reasoning.
This somewhat means that these models are trapped within the universe of "human capable reasoning" with some possibility of escaping it through the stochastic generative processes they're built on. But they simply can't think through novel problems and arrive at new conclusions.
Furthermore, they're limited by the fact they're built on human knowledge as encoded in text, which terribly imprecise and fluid. Any hope to have a path to AGI, where reasoning might actually happen, will have to have something far more rigorous for the internal reasoning, with language just being a clever interface rather than the mechanism by which the thinking is done in.
They're really "Large Analogy Engines" or maybe "Large Captured Reasoning Engines" and are an incredible technology.