[kaitai]

I teach a class that is about 80% Chinese students who have come to the US. Many have much better technical skills but I do see gaps in these concepts like graphs as the locus of points satisfying a logical claim, the equivalence of various approaches to integration (numerical/limit of Riemann sum/fundamental theorem of calculus), etc.

You're right that American textbooks on math are often very soft. We try to make things so understandable that they are impossible to understand. There's nothing to hold on to. I am occasionally guilty of this in my teaching, although it can be very effective with people who do understand the technical basics.