An aspect hinted at in some other threads is the reduction in available engine thrust at high altitude. Thus the SOP "powering through" the stall recovery, i.e. minimum loss of altitude, has a big compromise built in; You linger on the ragged edge of a stall for a LONG while.
Better to drop the nose to a) directly reduce AOA, and b) convert some potential energy to kinetic, to recover faster. (Adding thrust too, of course) The only remaining question is "How much do you drop the nose?"
If you start the video I posted at about 50:20 it talks about briefed recovery actions when testing. The "LOWER THE NOSE" is firmly emphasised, and the interesting point for me was that your stall margin decreased due to Mach effect as you accelerate. The point made was one must accept trading 2-3000' of altitude for speed in the recovery.