That said, you certainly do not need to master all kinds of advanced math to be able to handle this kind of counting. For instance expertise in differential equations will not prepare you to understand why a quicksort is on average better than a bubblesort, and why a mergesort has better performance guarantees than quicksort, even though its average performance is worse (if all memory is equally fast to access).

Indeed. One might even go so far as to say that what most programmers do does not require this math at all as one is mostly selecting from a set of prewritten libraries (and thus algorithms) whose properties are reasonable well known and documented. For instance those writing software that is heavily DB based probably will never consider what sort algorithms are in use as its handled by the DB. And those that do need to use sorting algorithms rarely need to self-determine the algorithms performance as some clever math type already did it and wrote it down in such a way that we can make educated choices without requiring the math skills. Of course if you dont understandthe underlying principles you end up making bad decisions.... (Like maybe there are circumstances where its better to use bubble sort, or where using splays is extremely inefficient....)