Julia, I have a BFA from a rescue who is also physically timid because of mobility/balance. Her ankles are locked and she probably has never flown because her chest muscles are really underdeveloped. I have a feeling she spent many years in a tiny cage.
She is really afraid of falling, to the point that she would lash out and make really aggressive growl/bite-you sounds if you wanted her to step up. It has gotten a lot better as she learns to trust me and I learn how to move her around. I'm trying to get her to the point of letting me touch her feet so maybe I could move a bit for therapy, but flapping exercise is out of the question as her grip is too weak and she can't fly, so is too scared. Walking looks painful and awkward but we try to do a bit. For some reason, hanging upside-down in her cage seems a favorite and comfortable position.
She will now quietly walk a short distance from anywhere to stepup for me every time, and we can walk around without me getting whapped by wings keeping balance. She has developed a great enjoyment of chewing boxes and hard woods. It has taken a few years.
Anyway, as you and Eliot learn each other I am sure his confidence will improve and you will naturally see ways you can help him. You are obviously great at reading body language. If he will do flapping without you making him feel unstable by moving your hand that would be nice, but don't make him worry you will make him unstable. That is very important that he always trust you. Put it on a cue on the cage or something.
Just trusting you and going for walks around the house will help his balance more than you might think. Our bodies are not all that stable a perch. It is really amazing to think of how complicated balancing while perched on a moving human is, but it is truly a big accomplishment! Starting with that is a great thing!
If he will use a boing or swing that might help too but I don't know if that is in his comfort zone yet.