Chain rule is a bad example. If you cannot remember the chain rule, you do not understand the theory at all. And if you do understand what a derivative is, the chain rule is trivial.
Yes, you’re right. Chain rule is easy and intuitive. Although I had bundled its name with the mess of integration methods and considered it forgotten.
Right, the chain rule is foundational. If you don't understand the chain rule, then related rates and lots of your real-world engineering applications of calculus would seem to require memorizing a crazy number of specific solution templates.
Though, if typical calculus courses don't spend the time to ensure a good understanding of the chain rule, that would explain why so many students seem to think calculus is a gigantic sea of random rules.