I'd correct you by saying that they'll memorize it for as long as it's useful for passing a test because the teacher wants them to solve something a particular way. They'll soon forget it. I learned long division in 3rd grade, relearned it in 6th grade, learned the algebraic polynomial version in 10th grade (then was shown/told synthetic division which is cooler)... and at this point I know I'd struggle doing any of those. I'd rather use a calculator and get back to doing whatever it was that required the answer. In these crazy times where even most elementary kids have phones with calculator capabilities there's no excuse to be parading these mental tricks as "math".
I do agree though that a proof of this would be much more interesting than the technique itself. If I have kids I intend to teach them math myself. I don't think I was introduced to the idea of formal proof until 8th grade geometry, and everyone hated those proofs since it was a bookkeeping exercise of writing down every algebraic step and what theorem/axiom allowed you to perform it.