Your sure the pulley is not turning on the shaft, right?.... My question? Is there a pin or something which makes the shaft on the pulley make the gears move? Is it worth opening and fixing???
Here's an exploded view of one of the transaxles used on that model. Looks like there might be an internal key. You should probably go to JD Parts and check the model and serial numbers to make sure you have the right picture. I just copied the first one I opened.
John Deere - Parts Catalog