A hydrostatic transmission will make a humming or a whiney noise when it experiences a load. Going up a hill shouldn't be enough to make it go to a halt. My guess is that the belt may be worn out or stretched. This can cause belt slipping without the burning. The belt may also not be taught enough and the tensioner pulley would need to be adjusted. The transmission should be fine. If there were to not have any fluid in them, they would simply burn up or seize.
Try a new belt.