> remembering the multiplication table is the antithesis of multiplying in ones head

But it's not! Because in order to be able to multiply numbers in your head efficiently you must remember the multiplication table.

No. By definition remembering an answer is not the same as doing.

Additionally my experience tells me otherwise.

I'm sorry but this seems patently wrong to me. There's only so much working stack space in your brain. If you're constantly having to think through multiple steps to multiply single digits then you're going to be at a serious disadvantage when you need to solve a bigger problem that involves more work than just single-digit multiplication.

No remembering the table is the ROM-table part. You need the algorithm part too.