I'm not a head-stepper either and when I think back on it many of the birds I've killed hardly flop UNTIL I grab a leg. Anyone else notice this?
BOOM! A good shot with a ton of pellets to brain + vertebrae incapacitates said bird so he's done (just doesn't know it yet). The first few moments when he's crumpled there, oxygen is still in his lungs and blood still courses through his veins but the wallop to his neuromuscular faculties has rendered him immobile. By the time I gather up my stuff and WALK to him, I can usually see that leg start to draw up and those feathers begin to slowly relax and "lay down."
Sometimes they go plum crazy at the shot but more often than not if I don't grab him for couple of minutes, there's a kick, a quiver or 3, and he just sorta slips off to that big'ol roost tree in the sky with nary a flop.