Yes, which is why I made sure to mention that they are semi-fictional systems.
Still, the intuitive system knows things from exposure: single digit multiplication tables or prime numbers less than 100. This is more neural network like. The executive system, if I’m using your terminology right, takes over when we get to larger multiplication problems like 38*64. It might go through a series of steps learned in elementary school and start making use of working memory to calculate the answer. This is still operating on a neural network but it’s conscious and we can describe the steps we used to get the answer. Of course the algorithm used was learned by a neural network and certain tasks in the calculation were done intuitively. Someone good at mental math might have techniques to do nearly the whole thing intuitively. But still, it can be useful to distinguish the intuitive/unconscious system in use when we are at cognitive ease from the executive/conscious system layered on top. Its usage can even be measured by heart rate and pupil dilation.
Most of the thinking that we do happens subconsciously. We typically know the answer to a question before we consciously think the answer, and any attempt to explain how we reached the answer is fundamentally flawed: Whatever model we think we have, that’s not really how we processed the information. We processed the information with a neural network — in a manner more similar to AI than the models we think we use.
I was originally pointing out that here when you say “most of the thinking” you separate thought into conscious and subconscious. This separation has been explored extensively in the book I mentioned which agrees with you pretty thoroughly. I think it could expand some of your ideas on the subject and give you some interesting insights on the psychology side of this.