My teachers tried to explain to me that they were trying to teach me the method, not the answer. I always thought I didn't need the method, if I could find the answer without it. Nowadays, I think that wasn't quite right.
Same for some methods I've since forgotten for solving polynomials. They were non-iterative algorithms that could work with any constants. They weren't faster than other methods, but you could fall back on them when you didn't have anything better.
That is what they were trying to teach me. General solutions that could work with anything. While, of course, it's valuable to be able to intuit shortcuts on the fly, if that's all you're capable of, there's a class of problems you simply can not solve. If they don't click in your head enough for you to come up with a specific shortcut, you wouldn't have a general one to fall back on.
I've always used computers, and didn't understand why my teacher got so upset when I programmed equations into my calculator. Like you'd think writing a program around a family of equations would develop experience with those equations, and I thought it was a good way to speed up the solving process. Needless to say, teacher did not agree, and quickly learned to ban my calculator on tests.
(technically, it was supposed to be banned anyway because it had a CAS... but I got it because 68k TI-BASIC was better than the Z80 TI-BASIC)
I still don't know the general solutions for most of the problems I work with nowadays, and I think that sucks. I think AI makes this way worse on a scale much greater than mathematics.
Sometimes I wish I could go back to school and catch up on all the basics I've been cheating around. I feel like my apparent intelligence is a mistake and I need to develop a better foundation.
Looks like I was the opposite to you in regards to calculators. I always preferred to do it the hard way and only used calculators for boring things that I knew by heart, but would take me longer to do by hand and be error prone.
But I didn't pursue a career where this type of stuff was any useful and forgot many of the things I could do easily in school, like integral and differential calculus.
As the saying goes, use it or lose it. No matter how foundational, you'll eventually forget all about it if you do not use it anymore. There are even cases of people that completely forgot their first language.
I'm optimistic about AI because it can have more knowledge than any given human in storage. Eventually we all kinda specialize around something. Even the generalists among mathematicians have little insight into medicine, rocket science or astrophysics. So we all become the hammers of the saying "if you are a hammer, everything looks like a nail". But when including AI we get a broader perspective.