Rot IS fungus and bacterial decomposition has already begun. The reason the babies are being produced is the plant knows it is dying and trying like heck to keep living. I would plant these babies in pots no larger than 3" in POTTING SOIL. Potting soil is sterilized but if you are using too much water then spores in the air will start the decomposition of your plant to include the roots.
What kind of soil did you use, what are your watering parameters, is there a layer of rock or gravel beneath the soil above the drainage hole?
Get rid of all that soil. Get rid of parent plant. It is a goner. Bleach the pot before using it again. I'd get new never used clay pots, wider than high and for the babies start SMALL. 3" no bigger! Do not transplant or try to get these babies growing in too large of a pot. Critical for all potted plants but even more critical for succulents and cactus.
Don't worry about adding sand. Plain potting soil, I'd even cover with a dome or plastic wrap with Popsicle sticks holding the plastic off those babies. Moist almost dry soil and with the plastic almost NO watering. No fertilizer. 3 inch diameter pots in clay that will last quite awhile with this plant before needing to be up potted. Oh, use bottled water when watering, do not use tap water.
Great drainage, POTTING SOIL NO GARDEN SOIL, no rock or gravel below soil and above drainage hole, shallow roots need a shallow pot (preferably clay), water very infrequently and do not fertilize until they've become acclimated to their new pots and soil! And even then I'd use Osmocote 14-14-14 ONLY once per year. No water holding gels or sponges or added fertilizer in the potting soil! Lots of light but no direct sunlight unless hardened off or used to having direct sunlight.