That makes sense. And actually I think I could re-derive the algorithm from that explanation.

Maybe they need to explain it in those terms. Or use pictures.

I prefer to think of subtraction as the inverse of addition, and division as the inverse of multiplication. You only need addition and multiplication if you're dealing with rationals.

I've never thought of division as repeated subtraction: it's the inverse of multiplication, or literal division as in I have a pile of beans that I divide equally between N piles.

I don't think you're wrong, just seems an odd way to think about it, and fails to account for any remainder.

The remainder is what's left after you've repeatedly subtracted, and can't subtract any more before going negative. (I wish that one of my teachers had said this to me, but they were too busy getting everybody to memorize the steps.)