Reusing and blackboxes do appear a lot in higher level mathematics. Indeed, the idea behind abstract algebra is to hide 'implementation' details. The concept of abstract data type in programming is similar to structures studied in algebra.

  It is common for mathematicians to rely on theorems as black boxes(ex: classification of surfaces) even without knowing the proof. Secondly, people can even write research papers without knowing how to work with some object covered in the paper, by working with collaborators who are experts on a different topic.

  It would be helpful to isolate the essence of calculus itself from the symbolic techniques, for ex to actually calculate integrals(especially  magical seeming substitutions and nontrivial factorizations) as many of these symbolic techniques will appear in different topics even outside calclus.

Here's a criterion for testing this core understanding calculus - Can somebody given a problem (say optimization, or finding volumes) convert it into a standard type of differentiation or integral, then use symbolic software like Mathematica to do the computation and then get the right answer. Often, calculus students memorize standard recipes for problems and get confused by a problem which is not hard symbolically, but requires some thought to set up correctly.