I am glad someone mentioned this. I kind of doubt the instructor is saying "no movement" of the arm or hand, so that there is no following of the mouth. Getting your elbow connected to your body is important, and thinking of your seat moving up to your hands, instead of you hands moving back and forth, will help to put the horse on the bit.
Riding with two whips is something that was sometimes done (and maybe still is) in classical riding instruction.
If they are unwilling to put you on a new horse, that's something to discuss. Your own assessment of your ability and readiness to move on may or may not be accurate. The instructor may think you still have something to learn from that horse.
But, I haven't done school lessons in absolute ages, so I am not familiar with what is "normal" in school horses and group lessons, so I would agree that looking around to at least find out if what you have is pretty typical might not hurt.
Tinyliny she did say that, and I wish it had been recorded.
I asked the day before if I could be put on a new horse. The next day, I arrived there a bit early and she was quite upset with me for wanting a new horse. I tried to explain simply and she had said some things that were probably not intended to be rude but came across that way.. but I just smiled and she picked one out for me.
The mare I rode was completely opposite to the horse I usually ride. He is VERY forward. She is not.. not at all. She was a challenge but I enjoyed her a lot. Her gaits were very lofty and she has a tendency to drop her left shoulder but I took the advice of some here on the forum, and was able to handle that and get her night and straightened up.
As per usual, no one came with so I have no video but I did have... contact! I kept it at the trot and the canter. I tried not to pump but instead let the horse take my hands away and it was nice and soft and our lateral work was awesome.
The other riding school hasn't gotten back to me yet, so I will give it another few weeks till I get back with them, mainly so I can give this new horse a chance.