I really don't think he needs to be taken down a level, he is a "better" (more stamina, better confo, more advanced horse) horse than the one I was riding before. It could be different at your barn, but at mine the faster horses are not riden by beginners. The more advanced people in the class ride the faster stronger horses. He is not rushing I would say, he is just fast, but it's a comfortable pace and he doesn't have that big of strides so it all works out.
I agree I shouldn't have to pull on him, but that's just the way he rides. All the lesson horse have their quarks, but it would be boring and I wouldn't learn anything if they were all the same.