Sorry to say but from what I can see I don't see any hope:(. There's always next time:).

Unfortunately these guys can't be beheaded. At least I've never seen it. Once they're gone they are gone.

Yeah, this guy rotted :(

The best way to prevent that next time, would be to improve the soil. At least 50% perlite and 50% cactus and succulent soil.

Thank you.

Any ideas where I went wrong?

I'm gonna guess your mix. Your mix looks like it is full of peat and peat is not good for succulents. It does one of two things. Peat either holds onto moisture or once dried out turns to concrete. Next time use the advice Mesembs has given you and mix some perlite in there. This will help tremendously. Perlite aerates and adds drainage to your soil.

If I can add when ever you repot your succulent get all the old soil off the roots. So the roots are clean of the old peat:).

I find these especially prone to rotting out well after they have been watered in the wrong soil. I lost two pots of Royal Flush, one showing a bit if rot in some heads, but no visible trouble with the others, another one with no rot showing at all. And no water from me either, for quite some time. One day they suddenly were soft at the base: goner. No trouble with the plain P. nelii.

How do you get the old soil off of the roots without destroying the roots?

Lots of patience:). A chopstick and your fingers work well. Usually I use water, but not for these guys.

I've had lithops and pleiospilos recover from situations like this. You might just want to sequester the good part from the existing mush and soil giving it a surgically clean cut just below the translucent part. You might want to lay it somewhere that water won't pool or stay too humid nor do you want to stress it in too direct light although some is good to help it harden off. Sometimes I even like to stow certain items away in the cinder block cells like the ones in your staging cause it helps with both the over spray and sun exposure issues as well. I know these things can rot just for looking at them the wrong way it seems. A fail safe rehabilitating mix should have a majority quarter inch(minus) perlite maybe with the fine dust removed. If you have a fine screen laying around you cold probably reuse the soil with the inert material and eliminate some of the fines unless you suspect pathogens. One more thing: don't be in too much of a hurry to introduce it to soil or moisture.