It DID, however, require a day of nastiness in the bathroom. Caulk is not non-toxic at all. It's harsh and has scary warnings all over it. But Ken and I cut out all the old stuff, cleaned it well, and re-caulked.
This, in combination with RELIGIOUS drying of the walls and joints, has all but eliminated the mildew issue. Even though Martha Stewart is gross in her own way, I just LOVE her Homekeeping Handbook. She tutored me through re-caulking, and her mini-lecture about wiping the shower down worked. She says something to the effect of this: "if you don't want to use toxic cleaning stuff, you have to keep your filth in check with maintenance." That makes sense. I am much more inclined to spend a minute wiping the shower walls if it will prevent me from "needing" to used something harsh to cut the mildew/scum/grime later. Wiping the walls was always sold to me as a time saver, and I never really could make myself do it. When sold as a toxin-reducer, you bet I will.
It hasn't eliminated my shower problems. Next task: finding a shower liner that is "green." Our current one is badly in need of replacement (it's beyond elbow-grease dirty, and 2 gussets are ripped). I loathe the idea of buying another sheet of PVC, just to throw it away later.