Deb, I'll start with your comment about an alpha horse never retreating in the herd. No they usually don't. Why? Because he is dealing with OTHER HORSES. The lower horses in the pecking order are following direction from ANOTHER HORSE. They trust the leadership. Now when a HUMAN, A PREDATOR steps in to the leadership role and gets into a situation where the horse is unconfident he INNATELY will not trust the human. Why? Because of that prey/predator barrier. All horses are born not to trust humans. So they aren't going to say, "Okay Mr. or Mrs. Predator, I trust you."
That's not how they are designed. But if we RETREAT and use approach and retreat to build confidence, that will help the horse trust us because that is the opposite of what most human predators do.
It blows the horse's mind because he thinks we will keep pushing him.
I would ask a horse to go across a steam very similarly to the way you do. Only 2 differences. One, I wouldn't talk (just a personal preference) and two I would not keep a steady leg on the horse. To a horse, that is a form of pressure. So you are basically telling the horse "Keep going, go forward, go forward" and if the horse is scared that destroys confidence at some level. So what I would do is point the horse toward the stream and ask him forward. If he went, I would release and allow him to walk. If at any time he stopped (due to unconfidence) I would wait until he showed me some sign of relaxation then ask him forward again. If he was sticky then I would back him up and start again. I would definitely not push him past this thresholds.
For the horse not moving when I ask him out on a circle, I would stand in front of the horse (not at his side), put some feel on the end of the rope and point in the direction I want him to go. If he didn't go I would lift my stick, then wiggle it, then touch him on the shoulder or on the neck with the end of my string. I do NOT aim for the hindquarters when asking the horse out on a circle, that just tells the horse to turn his butt away from you. You want the front half of the horse to leave (b/c then the rest of him will follow) so you direct the front of the horse away.
If he's in a stall and I ask him to move and he ignores me then I ask as soft as possible but then add pressure until he responds, then rub. I never get mean or aggressive about it. But then you have to ask yourself, "Is the horse ignoring me because he is snotty or is he defensive and is frozen b/c he is scared?" Each one takes a different strategy.