Well it isnt entirely true that you need to support a walking horse at all times to gait them. When they are first starting out, sure they do -but the goal is always to be able to loosen the reins and allow them to move freely, collected and moving off the hind and carry themselves. When I first started my mare, she had an identifiable flat walk and a step pace ...now it is very rare to see her go out of gait, even in the pasture as I took my time with her and went through the motions with her slowly. I used a snaffle bit to train her, never tied her head, used any wild devices. Just simply ask reward, correct.
My husband (who knows very little about teaching or correcting gait) rides both our horses, loose rein, neck reining, and they stay in gait for him. The best thing for teaching gait is to get on the trails and have the walk through all kinds of terrain. Up hill, down hill, through grass, hard packed dirt etc. Another tip is don't ask for any more speed than they can give you in gait. It is much the same as a speed walking human...they can only go so fast in a speed walk and when they can't go any faster because they havent got their muscling and training to go faster they end up running or skipping...
I wouldnt worry about the speed of the gaits for theraputic riding, as any well trained horse will be quiet...a dog walk and a flat walk are slower than a trot. But I would make sure that the horse is well trained, desensitized and quiet before I used it for those purposes.
:) good luck