I've read from mathematicians who consider their computer to be a tool, and from others who see the computer as a partner. As a hobbyist who studies automated theorem provers (like coq), there is so much elegance that goes into the computational proofs in order to even construct such a program that allows a mathematician to do their work.

Can you honestly say that it doesn't take an extreme amount of effort to construct the perfect typing system that allows you to even begin to be able to write a proof with a machine? People have to prove that theorem provers are correct with respect to that which what they define. This is no trivial task, it's an entirely different domain of knowledge.