[Question] What explanatory power does Kahneman’s System 2 possess?

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?