In short, the lack of a proof-based class renders students Calc-clueless. (groan)

This happened to me. I went straight from high school calc into college differential equations because the AP score allowed me. It took a Rudin-based analysis course, much later, for me to appreciate proofs of convergence or epsilon-delta arguments, because my H.S. calc did not have them, and the college diff-eq assumed you knew them already. The shock was painful.

Eventually though, you learn what you need to know.