"Just wash them with clean water on the slow cycles and an extra rinse if you feel you must"
That isn't the province of the old "wringer" washer that's described above, there ain't no cycles to it. These, you just fill with water and forge ahead. Rinsing is a separate operation that either requires draining water from the machine and refilling it with clean water or getting a separate rinse tub.
I had one last year, but found it was more trouble than it was worth. I had to keep it in a cold building and the water would freeze so I had to drain it every night. Also, it took quite a bit of water to get the machine full enough to use. I found that a 5-gallon bucket of cold water with a few drops of dish soap would do the trick. If the pelt was particularly soiled, I'd just let it soak for a few minutes. This would be followed by a rinse in a bucket of clean cold water.
However, the wringer on that machine is really handy. I dismantled the washer and mounted the wringer outside my shed so I can continue to use it to wring out wet pelts. I wash the pelt in soapy water, wring it, then wash it in clean water, and wring it again.
I should add that, with the exception of coyotes, I too try to get the critter cleaned up
before I skin it, and not after.
-- Hal