Imagine you had to look up every word you wanted to use, because you had a poor vocabulary. This would get old, fast.
The trouble is, many people don't have a genuine need to internalize computational technique as a way to see.
As a researcher, I find one of the main values of manual calculation is as error correction for reasoning, and a source of intuition. When I am in love with a certain idea, carrying out calculations related to this idea is a way to slow down and become objective about the idea and to test it from many angles. I am constantly guessing what should happen and checking this against the results of the calculation, which I attempt to do accurately and honestly to bolster or debunk my belief.
At some stage, when trying things by hand to develop intuition and generate ideas that will bypass computation, it may become appropriate to use technology to continue this process. Usually, though, moving to technology without having developing feelings by calculating many pages by hand feels very icky.
I'd like, as an educator, to try to motivate students to calculate for similar reasons to why researchers do. Unfortunately, I often don't do a terribly good job of tying "getting the right answer" to "debugging reasoning". There ends up being very little student metacognition or conjecture to be tested...only whether the number matches the answer key.
Incidentally, this is similar to why we ought to care about grammar and even spelling...for metacognitive reasons. Grammar, despite its conventional aspects, provides logical checksums that can rudder thought. Subtle differences in meaning can be detected by one's critical consideration of turns of phrase. Mindlessly relying on spell check or denying the need for grammar has similar detrimental effects to the blind use of Wolfram Alpha and calculators. Of course, these things are helpful, but can easily erode one's mental acuity if one is not very careful.
Perhaps the way to go is (1) Focus on reasoning about mathematics, and thinking about what it means; (2) Giving problems that require thought and ideas, and that would benefit from accurate exploratory calculation.
Standard Calculus often fails this miserably, I'm afraid.