The thing is, most fields that apply math do not need to "deeply understand" math. I think it is highly arrogant to imply that everyone needs a deep understanding of math to use the tools it provides. For most people which aren't mathematicians, it's perfectly fine to just be able to apply formula. They don't gain anything from a deep understanding of the matter.

It's the same as libraries in programming, really. You don't need to understand how a library works, you need to know what it does and how to use it. And in many cases, that's perfectly fine. After all, who would have the time to study the source code[1] of all libraries they use? You'd study that if you are interested, noticed a bug or need to know some specific detail of the implementation, not just because you need to use it.

[1] Not to mention those unfree binary libraries which you couldn't inspect even if you wanted.

A deep understanding allows one to deal with problems that they have not only been explicitly taught, but those they have also never encountered.

I don't think your example regarding libraries really applies well here. I think a more appropriate example would be knowing how to do a few things with a library without a real ( or any ) understanding of the language its based upon.

I'm no mathematician, nor am I a (professional) programmer for that matter, yet I've seen real benefits going back and really learning some of those topics that were kinda passed over in high school ( so what exactly is sine doing to my numbers? Taylor series? Fourier series? Optimization functions?). Besides, math teaches you to think, memorization doesn't.