I am a VERY anxious rider. I know for myself, I have to FEEL confident in what I am doing in order to get around that mental block. So start at the beginning. There is so much confidence that can be gained at simply mastering the walk and trot. I trotted for months gaining confidence. I am just now starting to canter my horse over jumps. Do lots of body building/strength building exercises as well. Just afew off of the top of my head are:
Walk and trot standing in the stirrups, 2pt in the stirrups. stand with no stirrups in the walk. Work different shapes around the arena in the walk and trot focusing on using the body and not just her reins. Do that with no stirrups as well. Sitting trot with and without stirrups.
Those are just a few.
I have to say, I don't think your BO handled that well at all. From someone who is terrified of most things, going from a crossrail to cantering a 2' oxer would send me into hysterics and I would have left. There is a time to give a good nudge and I personally don't think that was it.
You make a wonderful point, and it's nice to hear from the student's point of view the same situation. You're right, when someone is afraid, adding pressure Never helps (same with training horses ;) )
I think you're idea of spending time working on strength and confidence building exercises is wonderful. With a number of my recreational riders I see other trainers just throwing them in front of a jump before they even fully understand the mechanics of riding, still steering with all hands, not understanding or feeling the horse's stride.
I think some no-stirrup lessons (of course starting slow) would do huge for her confidence too! I also think the student should learn some ground work skills too, working on controlling the horse from the ground will give her a stronger feeling of control, control=less fear. Have her work on a lot of transitions, if she knows she can stop confidently, smoothly and comfortably, she'll be less worried about trying higher speeds. Give her time to learn the feel of the horse, learning how to manipulate different aspects of the horse, such as disengaging the hind end, steering with seat+legs rather than hands. All of this will give her a better understanding of how to the horse moves and works and with understanding comes also less fear.
This student sounds like she'll be a lot of fun for you to work with :) It was nice to hear about your experience as a nervous rider too!