... but I'd say it's pretty obvious (both in introspection and in observation of others) that people don't naturally use formal logic for anything ...

Yes. But some place too much confidence in how "rational" their intuition is, including some of the most intelligent minds the world has seen.

Specifically, many operate as if their intuition (that they treat as completely rational) has some kind of supernatural/magic/divine origin, including many who (imo) SHOULD know better.

While I think (like you do) that this intuition has a lot in common with LLM's and other NN architectures than pure logic, or even the scientific method.