I guess I've rarely had situations where I had to manually execute most of that machinery. I agree the rote machinery is quite useful to apply, but the question for me usually is what to apply and how. Applying it is a job for computers.
For example, it's quite important for me to know when I might need to compute an integral, and what I'd do with it. But when it comes to banging out the symbolic manipulations, remembering tables of what integrates to what, recalling integration strategies for common kinds of integrals, etc., Mathematica is more skilled than I am, so I defer to its expertise, and haven't tried to keep any of that stuff in my head since high school.
One case I have trouble doing in Mathematica, etc. are variable transformations to increase numerical stability. A lot of times if you're clever you can do a variable change on an improper integral with (integrable) singularities to get an integral of a continuous, bounded, function over (0, 1). Working out this change of variables is something I find easier to do by hand.
I also generally have trouble getting Mathematica to do all of the work I need to do to come up with error bounds on numerical schemes.
For example, it's quite important for me to know when I might need to compute an integral, and what I'd do with it. But when it comes to banging out the symbolic manipulations, remembering tables of what integrates to what, recalling integration strategies for common kinds of integrals, etc., Mathematica is more skilled than I am, so I defer to its expertise, and haven't tried to keep any of that stuff in my head since high school.