Using Oxiclean powder, I made a thick paste, applied it with an old toothbrush, and let it sit overnight. Then used an old electric toothbrush to remove the paste. Worked well.
Also tried sprinkling a heavy layer of baking soda then spraying with vinegar so it bubbled. Left it on for several hours then removed with a toothbrush. Worked well enough to be a feasible alternative to the Oxiclean method.
Several points to keep in mind when reading this answer:
Used it for grout; no vinyl or silicone edges involved.
Both methods made the floor slick; had to rinse and wipe several times to "unslick" it.
No reason to think the pink stuff was mold. When touched, it felt slimy. My guess is that it's a combination of soaps, shampoos, and as mentioned in another reply, flouride.
This is a cleaning solution. The holy grail would be the clean-and-prevent solution sought in the original question.