I'm pulling schematics from a 1999 model, I figure your mower is about 10 years old from what you have said.
The transmission is a probably over $100. I would suggest pushing it!
If you wanted to, you could drop the trans and see if you can get it apart but I do not think parts are available for the internals. Possibly it's just something stupid. Looks like there's a lever on the box that may?? engage it?? Are you sure it was engaged when you were testing it by hand? Looks like there's a cable going to a clutch or the box.
I'll be honest here, I do not work on mowers but know my way around a lot of power equipment and I stayed in a Holiday Inn last night. Here's a pic of the box I was looking at. Tell me if it's close to yours.
Let me know what else I can help with.