I 100 % beleive, and KNOW from experiance that you can indeed train a dog using a Harness. I have a yorkishire terrier, and she wears a collar, but only to hold her tags in case she goes missing. If she is tethered for a walk or training, she is wearing a harness. And for a 9 month old puppy, she is very very well trained and listens to me very well. And she is lead trained, which can be done without the use of a collar or a leash.
It bothers me that there is a yorkie in your class being forced to wear a collar. Yorkies are proned to Tracheal collapsation. Its bad enough when people use regular collars on them, so i really hope it isnt a slip collar or anything that is being used on the poor dog!