For a lot of math departments, that is exactly why they teach this. Education is rooted in application. We have entire careers that depend on certain aspects of mathematics, so most companies gatekeep that career by a degree. The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications. Knowing and not telling them is doing them a disservice.
Depends on the course. That's why some departments have separate calculus courses for math majors - because otherwise the whole class will be full of non-math majors (engineers, etc) and focusing on their needs does a disservice to the students in their own department.
> The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications.
If I'm a CS major, and the degree is requiring a class outside of the CS department, you shouldn't expect the professor of the class to know why the CS department is requiring it. It's on the CS department and its faculty to explain it.
Even in more applied fields it can take decades before advances become practically relevant. Restricting teaching to topics that have immediate practical relevance would therefore do students a huge disservice and prevent them from approaching the knowledge frontier of the field.
The head of my CS department was quoted as saying, "our job is to produce researchers!"
This was in the context of a conversation about why the program wasn't better suited for industry.
It turned out that in our department 7% of students ended up going into research.
So he was openly declaring his intention to disregard the needs of 93% of his students.
(I'm not arguing against research, or on going deep with the theory. I just thought that was kind of funny, and it seemed a bit wrong to me.)
Other engineering departments are not as fussy about the distinction.
See also: Using spaced repetition to see through a piece of mathematics
https://news.ycombinator.com/item?id=18895613
Where the author describes his mind eventually turning the mathematical objects (symbols) into some kind of mental objects he was able to actually manipulate "directly". As a result of this facility, his mind spontaneously produced many new insights into the matter.
That level of fluency obviously requires a great deal of work, but in my experience it can be approximated and accelerated with analogies. (e.g. see Feynman's teaching.)