Very good points, I think fundamental to this is the reliance on the weird mixed theory/application version of calculus that is taught in our institutions: a gr

Very good points, I think fundamental to this is the reliance on the weird mixed theory/application version of calculus that is taught in our institutions: a great deal of it is generally just developing a bag of tricks to help you find analytic solutions for integrals of elementary functions. The reliance on calculus at all is I think because of this fact: calculus provides a huge number of problems which are easy to design and can be made easy or difficult to solve depending on the arbitrary tricks they require, whereas a theoretical approach would require fundamental proof-based analysis questions which are far more difficult to both design and solve. And a numerical approach while extremely informative useful and natural isn't suitable as a "mathematics IQ test" weed out course that many people seem to want calculus to be. Indeed the math subject GRE was (when I took it) primarily many, many tricky calculus questions with a very strict time limit, because these problems are fairly straightforward to write and standard.