We're not at odds! I think all I'm saying is that when manually working out integrals gets frustrating, there's a learning-mode escape hatch to just figuring out how to solve them in something like Sage; when the frustration subsides you go just get back to doing the manual stuff. When I'm in Sage I'm still learning stuff. I'm never abandoning analytical work.

I would feel real weird if there were things (in Calc II problem sets) I could solve in Sage that I simply couldn't do by hand.

I don't feel weird that there are things I can do, but will get wrong a bunch of times if I try to do them, and can quickly bang out in Sage. That seems fine to me. For a lot of these subjects, I don't care about automaticity, just intuition.

It's like a lot of linear algebra: being able to quickly do things by hand is kind of silly, because for real world problems (at least in data science) hand solves aren't even really feasible. But learning to do it by hand is important for building intuition.