I think it's important to distinguish reinventing the wheel for practice or understanding, and reinventing the wheel out of stubbornness or "not built here". The first is important because, as you said, it really helps you understand some things that might be rather opaque initially. I didn't understand backpropagation until I implemented it myself.
But what you don't want to do is reinvent the wheel just because you can. Don't write your own crypto, and use whatever FOSS software (with a license approved by your employer) and existing solutions that you can, unless you really need a feature that they don't have. Programming is awesome in part because we can build solutions on top of other pre-exsting solutions; it's 2017 and I can code Python in a web browser with interspersed markdown.
But what you don't want to do is reinvent the wheel just because you can. Don't write your own crypto, and use whatever FOSS software (with a license approved by your employer) and existing solutions that you can, unless you really need a feature that they don't have. Programming is awesome in part because we can build solutions on top of other pre-exsting solutions; it's 2017 and I can code Python in a web browser with interspersed markdown.