...I have an inkling that my instructor puts me on more stubborn horses because she thinks that's where I need to be - working on my dominance. I tend to want to treat the horse like a dog or one of my cats; like a buddy. This sort of approach doesn't seem to get the job done in the saddle...
A horse that doesn't want to move will not help you learn dominance - which would be better described as having the confidence to convince the horse to accept your decisions.
My last lesson (#4, although I've been riding my own horses for 2+ years), I was put on a horse that DID NOT want to go. I have a very dominant (domineering?) personality and enough confidence, but the horse just didn't want to go. 20-25 kicks - and I mean as hard as I could give them - were needed to get a slow trot, and he'd slow to a walk in 50 yards.
So the instructor brought out a crop. "You shouldn't have to beat yourself to death making him move...so ask with a tisk, squeeze, then bump once - and then a light smack on his hip with this. If one light smack doesn't do it, give a hard one."
Meanwhile, the horse was looking at the crop like it was a snake. So I took it, asked, squeezed, bumped - then an firm tap. At the tap, he trotted. If he tried to slow, I'd squeeze...then tap if needed. After about 10 minutes, he performed the rest of the lesson moving out with a light squeeze.
With my own horses, the problem is almost always slowing them down. The lesson I took away was that some horses will NOT respond properly to a squeeze or bump (or 20 hard kicks) unless they understand that you have the option of asking less politely.