As I recall Kahneman is somewhat careful to avoid presenting S1/S2 as part of a dual process theory, and in doing so naturally cuts off some of the chance to turn around and use S2 causally upstream of the things he describes. I think you are correctly seeing the Kahneman is very careful in how he writes, such that S1/S2 are not gears in his model so much as post hoc patterns that act as nice referents to, in his model, isolated behaviors that share certain traits without having to propose a unifying causal mechanism.
Nonetheless, I think we can identify S2 roughly with the neocortex and S1 roughly with the rest of the brain, and understand S1/S2 behaviors as those primarily driven by activity in those parts of the brain. Kahneman just is careful, in my recollection, to avoid saying things like that because there's no hard proof for it, just inference.
In the 70s and 80s, Kahneman and Tversky did a bunch of pioneering research on heuristics and biases in human thought. Then, in Thinking Fast and Slow, Kahneman divided human cognition into System 1 and System 2 - basically, System 1 applies quick heuristics which are prone to biases, and System 2 does the slow, effortful thinking.
But what does System 2 actually add to the theory in terms of explanatory power? Consider an alternative version of Thinking Fast and Slow in which Kahneman wrote something like "Here are the conditions in which humans use this mode of reasoning I'm calling System 1, which is fast and approximate and effortless and uses heuristics and demonstrates biases which can be detected in certain ways. The rest of the time, I have no idea what's going on, except that it doesn't display the traits that would qualify it as System 1 inference." In what ways would this be less informative than his actual claims?