Okay, I only watched some trotting for now since I'm at work and taking a short surfing break. What I see, over and over, is this horse being pulled in from the front with overly busy hands. When the rider passed by very close to the camera, you can see her elbows coming back with each posting motion instead of just opening/closing and not moving forward or back. The horse going around BTV confirms the same. I see lots of circles, but not attempts at half halts or transitions within the gait to help the horse engage it's hind end.
The horse has already learned to duck contact, so the only way to fix it is to retrain it to stretch forward into contact, not try to hold it into a frame. If I were on it, I would be letting the reins out a few inches to allow the horse to stretch, doing lots of half halts, spiral in/out circles, and transitions within the trot to get this horse out of the habit of ducking the contact and engaging it's hind end. If this is one of those horses who loves to use the rider's hands to hold it up, I would go even further and trot along on a loose rein doing big sweeping circles to start getting the body working more and take away the focus on the head.
Will watch canter later.
You just have to see your distance...you don't have to like it.