I agree math shouldn't seem like a magical black box, because it's anything but that. I think anyone who uses mathematics extensively should know about their inner wheelworks.
But...
Should electric engineers use matlab instead of doing fourier transform by hand?
I believe it's much more important to understand why fourrier transform works than being proficient at doing them by hand. And I mean having gone through the whole bunch of theorems and their demonstrations (which are about why the theorem is true more than persuade you it is) on the subject, and doing a few of them by hand, to get a feel.
During my three years undergrad physics, we had all our math course with the math students, and we were asked the same (sometimes sickening) level of rigour as them. Sometimes I felt it was too much. But now, when I have matlab compute, say, a fourrier transform, to reuse your example, I exactly know what's going on and why this can be done at all. I think this is of prime importance when doing science, more than being able to perform it manually.
My point being, once you really know what you're doing, why don't let the machine do the dirty work ?