Peter;
Thank you for your thoughtful post.
Yes, I not only agree with your "notwithstanding", but know of at least one reasonable explanation for the immediate aft-stick movement.
Still, the key here is in the notion of "stabilizing the aircraft". One must absolutely establish control of the aircraft and one does what one must to maintain control, which also means avoiding those actions and changes which will de-stabilize the aircraft. So, for example, if there is a momentary apparent "loss of altitude", say 300ft (I am not citing a known example, I'm positing, here), the reaction could, (not 'would'), be to pull back and "regain" the altitude, from which one then levels off, sets power and gets out the QRH. I'm using quotation marks in this example because any such loss would almost certainly be an indication issue and not an actual loss of height, and in any case, such loss, real or merely indicated, is not serious and does not require instant response or correction when one is wrestling with another abnormality. Sure is easy to say and type, isn't it? My golly. We have to be so careful of hindsight bias when reading what is coming.
Of course, I, we, don't know if any of this occurred, but the notion does lie within what I would consider the "plausible" arena vice some of the theories placed out in the open for consideration by other contributors.