I would argue that memorization is important, but what you memorize and how you arrive at that is very personal. Forcing kids to memorize very specific things, and trying to enforce memorization (as opposed to the ability to arrive at the solution) seems like a bad idea to me.

I still don't have the 10x10 multiplication table memorized, and I took physics in high school and work as a programmer. I have a use for knowing number multiples, and have domain-specific numbers memorized (2^8=8*8=256, 256*256=65536), but what I don't remember off the top of my head I can figure out from the things I do know, from certain tricks, and from brute force mental math juggling numbers.

And the important thing to me is, I learned what I know not because somebody told me this is how I should do things, but because I picked them up as needed, a mix of memorizing common multiplications and figuring out tricks (like multiples of 9*N for N<11 being the digits N-1 and 10-N)

I strongly disagree that memorization is important or foundational to advanced math. It definitely is useful, but you don't need it. And the more advanced your math gets, the less valuable it becomes.

My experience is that university-level math explicitly tells you to not memorize values and formulas, but to get comfortable finding solutions directly, because then you actually learn what is going on and have methods that are universally useful.

In the real world memorization is even less useful. You will never be as fast and accurate as a calculator, or remember as many values as a precomputed table has. So why bother?