So - I had no luck finding any replacement brake pads for my AMT 626. There's no way I was going to pay $156 for 4 tiny brake pads from the local Deere dealer - that is just plain excessive.
I only needed new "stationary" pads - the pucks on the actual moveable piston side of the brake assembly still had plenty of material left on them'
After a bunch of reading all over the internet, I finally realized that re-lining a braking system used to be a common maintenance item, and wasn't at all difficult to do. I could do it myself.
So - I had the old backing plates. I used a small chisel to remove what was left of the old brake material, and sanded the rest off. I roughed up the surface with very coarse sandpaper.
I purchased some 3/16" wire-backed brake lining material off eBay, for roughly $5. Used JB weld to bond it to the old backing plates. Put them in a vise overnight to cure. After the JB Weld cured, I trimmed the edges of the lining with sheet metal snips until it roughly matched the profile of the backing plate. Installed them in the brake calipers, and they work just fine! I carefully tested the brakes to see if I could get them to fail, debond, etc. by trying really hard braking, panic stops, dragging the brake pedal to get them really, really hot - and so far, they seem to be working without trouble. The only thing I have any concern about is that JB Weld is only good to about 500° F, and I haven't been able to get them that hot, even with abuse - but I don't work this Gator very hard. Someone else may need a more heat resistant adhesive, and it's available - but I had JB Weld handy.
While I had the brakes apart, I took the shoes to a local metal shop and had him make me a spare set of backing plates from 1/8" mild steel. A set of stationary backing plates, and a set of the piston-side backing plates. Should I need to do this again, I have a set to play with. This cost me $10.
This was an easy fix, if I can do it, anyone can! Hope my experience helps out others.