One half of the fleece is still soaking in a tub in my room. The other half I decided to try the washing machine method with. I have to say it's cleaner than the fleece in the bucket, but it really ought to be given that it's had 8, yes count them 8 washes. It's still kinda poopy, but I've resigned myself to just picking around it, and combing it out once dried. When I soak the finished yarn to set the twist I'll add soap and give it one more wash to get the residues off, but any solid matter is just gonna have to be manually removed at this point.
I think the fleece in the tub is just going to get a couple rinse cycles in the washing machine, spun out well, and then it's on to drying and carding the whole lot.
While I think it's the number of washings and not the method that got the machine fleece cleaner, I must admit that it's far easier on the back to let the machine fill and drain the tub. Spinning the fleece also removes most of the water, far better than I possibly could have by hand.