A quick update since I'm behind on all of my machine threads!
This washer has been back together and running smoothly for several months now.
Since the agitator shaft gumming up was what killed this transmission, we had no choice but to press out the pin on the drive gear and remove the shaft to clean it. The general advice is to not attempt this on a working transmission, and after doing it ourselves, I completely understand why. That groove pin is in there incredibly tight, and the force required to drive it out in our case fractured the pin.
To our surprise, that groove pin is/was one of the hardest to find replacement parts for this project. Nobody had them, and it took weeks to locate one. The seller had two, we bought both. Upon reinstalling, the first replacement pin broke from the force. The second one we chucked in the drill press, and filed it down a bit before installing, and it still took an incredible amount of force to reinstall. It's not going anywhere. We got very lucky there.
Our second stroke of luck was finding a NOS agitator shaft. I'd been searching for one for months, they just hadn't come up, nor has one been available since. Our old shaft could have been re-used with some work, which we were prepared to do, but this find simplified things considerably. A *great* repair job on a shaft can be seen at this thread, by user 'mit634'
https://www.automaticwasher.org/cgi-bin/TD/TD-VIEWTHREAD.cgi?77633
Lastly, my mother located what might have been the last available NOS pitman transmission gear set on the planet, and after a long journey from Australia, they arrived. The timing was such that all of this happened while the original gears were still in the machine shop, so they were purchased to have as a backup plan in case the originals were unrepairable. In the end, it is the new set that made it back into the transmission, with the repaired gears being set aside as spares. The NOS set had 30-40 years of "shelf wear", banging around from transport and their long journey across the globe, so a few dings on the teeth were filed down to avoid any uneven wear once put into service.
Everything else was relatively uneventful. Upon reassembly, we counted teeth and re-oriented the agitator shaft gear so that the broken tooth was away from the segment gear, and would never see service since the agitator doesn't complete a full rotation. (This is easily verified by looking at the wear patterns on the gear from prior use.)
Mom says her A308 now starts quicker and runs smoother than it ever has.
Ready for the next 40 years!