You might be able to fix it without splitting the tractor. First, open the access panel on the driver's LHS where the fenders meet the frame. Using an air hose, blow out any loose debris that you can from that vantage point. Then, get some degreaser in a spray bottle and soak it down, rinse off, and then use the air hose again to accelerate the drying. When you have cleaned it to your satisfaction, then you can spray it with some lubricant (Like PB Blaster). Spray the nipples on top of the valve and soak them well with the lubricant. After allowing them to soak for a while, maybe a day or two or three after repeat spraying of lubricant, use a long shaft screw driver (or I prefer to use a piece of wood--kind of like a paint stir stick) and try to depress the nipple on the valve down. Once the lubricant does its thing, they should work freely and your reverse should return to your tractor, if the rest of the linkage is in order. It is worth a try before splitting the tractor, as that is quite a bit of work.
Hope that helps!!
Cub Cadet 123
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