So, in one sense I agree with you. There is zero evidence that the human brain runs separate systems for separate types of cognition.

On the other hand, the reason this idea is sticky is because it matches our conscious experience. In some situations, we respond intuitively. In other situations, we choose to work analytically using tools like research, deliberation, note-taking, etc.

I think it's this second sense in which people are using the term with respect to LLMs. And it's not a terrible analogy.

However, comparing "neural networks" to actual neurons is almost never useful.