When in an 'emergency' situation, like a horse going to buck, or bolt, or a spook type incident, I do as the OP has suggested...you just bring the rein in to get the horse to bring his head around and hopefully gain some semblence of control of his feet within a few strides. And if your training in the controlled areas has been successful that "should" happen.
In your situation, I wouldn't necessarily have been focused on the ORS as just getting his feet unstuck so you could get him 'thinking' again, rather than reacting. Not necessarily moving toward the steer, just moving his feet in circles, or 8s, whatever.
"The ideal horseman has the courage of a lion, the patience of a saint, and the hands of a woman..."