Off the top of my head, for just mathematics:
- Long division
- Calculate limits
- Curve sketching
These aren't things I do by hand any more. But if I hadn't learned how to do them, I wouldn't be able to apply them to more advanced problems.
From my extremely limited exposure to US education methods, I would concur that some algorithms are practiced in an excessive manner. But that does not mean that they are useless, even if computers can help you often enough.
Maybe you can name a thing or two that you consider "busywork [that] nobody does any more because we have computers and calculators"?
(I will note though that biology courses always stood out to me in exactly the way that is lamented in this thread - almost entirely memorizing arbitrary things.)