Something tells me it's just the pulleys, belts and maybe a gear on the deck. If it were me, I'd ask them for a run down of what's wrong, with a list of the parts involved, then drop that deck at home and do your own repairs. Worst case if the drive shaft runs then you know it's the deck and you can toss the deck in a truck or back of an SUV to get to worked on. 10 years ago they were already making those things to where it intentionally throws a pulley or belt or gear to avoid straining and wiping out the engine. As you say, the mower runs great, so that tells me it's only the deck.
+1. It could be the bearings and shaft, too. I've never worked on a John Deere mower, but most of the time dropping the deck isn't difficult, nor is changing the deck bearings. Considering the cost of labor, the parts should be less than $500.
Coincidentally, I'm looking for a new mower because I can't find the necessary parts for my 12 year old mower and it's getting pretty worn out. I was toying with the idea of a John Deere in the hopes the parts would be available longer.