That again seems to be presenting a false choice, though -- that because this is a good thing, memorization is a bad thing. I think that this is a good thing AND memorization has its uses.
I think that what you describe is great way to learn multiplication, and that's how my kid learned it too (at least part of the way, and how she was introduced to it at the earliest stages).
Sure. I have no problem with this. If my daughter decides she wants to make flash cards for herself, I will go along as long as it doesn't interfere with her understanding.
What makes me bristle is the idea that so-called "basic skills" (meaning rote memorization and mechanical ways to solve problems of a very specific format) are necessary.
From personal experience I know that memorizing the multiplication table is not that important even for someone like me in an engineering career. When I see 6 x 8 my mind takes a little trip through 12 and 24. This means that your kid might be able to solve this problem twice as fast as I, maybe in .15 second where it takes me .3 seconds. This has really never been much of an impediment (or I would take the time to learn them now).
But my thesis is this,
At the core of mathematics are expressions and objects and ideas. The goal is to develop and express new ideas using creative thinking and logic and problem solving.
You clearly don't need FOIL or BEDMAS or silly tricks or rote memorization to get to the point you can do real mathematics. I didn't need them. My daughter is doing real math without any such tricks.
I will accept the idea that these tricks are helpful to some people. My objection is when these tricks become the focus of mathematics. In my opinion it is essential to make the ideas focus of mathematics. And, thus, I feel it is best to start with the ideas.
That being said, you are correct about different people learning differently.