Look at the video again and study the rider’s hands. There’s almost always a straight line from the horse’s mouth to her hands. The horse is never behind the vertical (his head never seems to move from vertical to behind the vertical line showing compression and tension in the neck). Instructors try to get us there by asking us to visualize breathing through the reins. If she was pulling and tugging like you though, you’d see him (the horse) run into a brick wall every time he stretched because her hands wouldn’t give. This wasn’t the case here.
Finally, this is a lot of horse. He’s a gymnast, an acrobat, and a body builder all at once. Look at his muscle at the end of the piece, when he’s walking on a long rein. Look at how they cantered down the long sides. So her job of holding him (the knife’s edge analogy) is huge. She is holding a lot of horse.
This is a video for study. Thank you for posting it.