You're correct, but in the same way that you'd be correct in saying you can derive most of modern mathematics from the axioms of Zermelo–Fraenkel set theory. T

You're correct, but in the same way that you'd be correct in saying you can derive most of modern mathematics from the axioms of Zermelo–Fraenkel set theory. Textbooks still go through the individual results rather than just stopping at an enumeration of the axioms. I think the first three points get to the main body of the actual errors many people make in the exact way they make them: overconfidence in their understanding and in their ability to be neutral.