Excellent work to find the Model and Serial number plate. Thank you.
Based on this, your machine is the 1984 Groundsmaster 72. Actual Model is the first set of numbers, being 30780 and the serial number 40315 shows it was made in 1984, being serial numbers 40001-49999.
I have checked the Toro website (link below), and confirmed the parts listings indicate this has the Renault R839-46-15-2005-A engine and the Transmission as being 48-1790, which you mentioned earlier.

https://www.toro.com/en/parts/partdetails/?id=15708

From this link (above), you will be able to get both the Operators Manual and Parts Manual. Both of these are 52 pages and are available to download Free from the Toro website. Link to the Operators Manual for your machine is shown below: -

https://www.toro.com/getpub/7604

Pages 32-34 of the Owners Manual deal with Traction Drive Maintenance, which your machine may only need some adjustment to fix the issue around lack of forward motion, and Pages 39-48 details Hydraulic System Maintenance, including a Troubleshooting section on Page 42 that could help determine cause.

Let us know how you get on after looking over the manuals as it may help others in the future.

Hope this information is of benefit and cures your machine.

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