Actually, most algebraic tricks in calculus, which are what make it challenging, are pretty unnecessary to know if you can program a computer or use existing math software. Conceptual tricks, on the other hand, which might be more aptly called "conceptual applications," should be understood and derived. The unfortunate thing is that many students get so accustomed to memorizing tricks that would indeed require really advanced math abilities and lots of time to derive that they gloss over valuable concepts that they should actually understand very well.

No, not actually.

I only used the word "trick" because you did. There are NO TRICKS in math. If a practitioner uses the word "trick" its because he's being facetious or joking. The algebraic manipulations which you find challenging are absolutely essential to master inside-out to be able to follow more advanced mathematical reasoning. You'll see these "tricks" again in a profoundly more generalized form if you study Algebra again (abstract algebra, that is).

There is utility in doing algebra on your computer when you're dealing with literally pages for one expression. That is done to save time and reduce the probability of errors-- and NOT because you "can't" otherwise do it. Even then, you'll need to manually sanity-check the work using skills you learned doing all those tedious problem sets.

Memorizing culture is a social issue and an issue about teachers, not about curriculum. Or you think a teacher can't tell the difference between a student that memorized a few formulas and the student that understood the concepts behind those?

EDIT: When I was in high school there were a few teachers that frowned upon memorizing things. They would give low grades to such students. Then halfway my education I went to live in another country where memorization was the accepted learning method and I've felt the difference greatly so I know what you're talking about.