I don't fully understand the question, but let me say a few things roughly related to the question and hope this is useful.
In my opinion, one of the things that draws me to mathematics is understanding the hidden relationships between structures. For me, the most interesting part of a proof is that it can show you why something is true. If the proof doesn't help you understand why it is true, then either you don't understand the proof or the proof is unsatisfying -- either of these mean that more work is needed. I remember attending a lecture by Gromov in which he said, no one knows why Gauss's reciprocity law is true. A very precocious undergraduate student raised his hand and proceeded to give a proof. Gromov smiled and said, yes, I know many proofs, but I still don't know why it is true.
A calculation can also be similar. There are some calculations that tell you absolutely nothing. There are other calculations that are enlightening. Sometimes the skills you build in doing calculations that a computer could have done enable you to do calculations the computer could not have done... or maybe allow you to see a pattern that you would not have seen otherwise. Other times, you are just wasting your time doing straightforward computations and it would have been better if you had delegated the work to the computer. For instance, in my work, I usually have the computer find eigenvectors of matrices, do all plots, and integrate anything I need to integrate. The semesters I teach a class that does one of these by hand, my skills in that area dramatically increase.
Finally, there is a lot of merit to being technically proficient. All the greats of the past were monsters at computation (Newton, Gauss, Riemann, ...) Allegedly, Serge Lang would give his Calculus students a quiz in basic high school algebra at the beginning of the semester. Those who could solve the problems immediately, he predicted would do well; those who had to think, he predicted would not. (Here is the only reference I could find for this story, http://www.joelonsoftware.com/articles/GuerrillaInterviewing3.html). This matches my experience, to a certain degree, though I have no explanation as to why this might be true.
My conclusion then is that (1) it definitely makes sense to delegate some work to a computer, at least once one has mastered the concept, (2) mathematics, as I understand it, cannot be done by computer entirely, and (3) developing some proficiency at computation is essential so that you are able to understand the steps involved.