I am a new GH owner (623T) and just joined this forum, but as I understand it, all T series grasshoppers have a dual hydraulic pump that is directly shaft coupled to the engine, so there is no belt involved in powering the transmission. The pump feeds the wheel motors via direct hydraulic hose connections (visible under the seat). The two belts at the rear of the engine are indeed for the PTO, but they should not affect problems with movement - they would only provide power to the mower deck or other attachment.
If the engine starts and runs, then the pump should be getting power. If the pump is being powered then a problem in a control arm linkage, or a loose bypass bolt on the side of the pump body, might keep one wheel from turning, but with either of those causes it would be strange to have both wheels affected at the same time. The only thing that occurs to me as a cause for both wheels inoperative would be if the shaft coupling is slipping or broken? As noted I'm no expert, so I'd be inclined to go to the GH website and use their Ask a Technician service... Hope this helps.