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Offloading computation only works when you understand what the computations are and why we do them. That's something that must be learned, it's not knowledge that springs fully formed into our minds as soon as we step into a classroom.
Carrying out computations thus gives us explicit and implicit knowledge of how the things we may eventually automate actually work. But it's also valuable because it trains us to compute in a precise and effective manner—a capability that remains useful later on. For instance, in logic it's often important to be able to carry out syntactic manipulations (e.g. into normal forms) in one's head, or even tacitly.
I'm sure there are plenty of examples from other areas of mathematics where computation is important, it's just that we do it so automatically that we don't think about it. Often I've found that students have trouble following proofs that take logically and computationally innocent steps without saying what's going on. Here I don't mean things like applying AC in the background, but just simple tricks like de Morgan's laws or taking the contrapositive. They have difficulty because they haven't taken those steps often enough themselves to have internalised them.