The method you promote is the common way taught by many trainers, including me, until I found out differently. The walk, walk, head down, approach will not work for some horses. What I found out, just this year, works for nearly all of them. And works much faster. I saw horses that had 5 month of professional training under the walk, walk method do nothing but trot. I had given up on them until I happened on to these trainers. In one afternoon with the head up drowning method they started gaiting.
Like I said. This old dog is not too old to learn new tricks. I had been using the walk, walk, method too, but found out there is a better way. It will not hollow the back, and it doesn't take but days of short rides like this, before you can start slowing them down and bringing the nose in a little bit at a time. We did in weeks, what could not be done in months with the walk, walk method.
Better try it, before you condemn it, an claim it's false. The walk, walk, way works on some, but usually takes a very long time, and will not work, at all, on some.
I little food for thought. When you first start a horse moving, under saddle, at a gait above a walk, and don't restrict the head, where does the head go? So what is more natural the head up, or the forced head down?