According to this article from Curd Nerd
"But it’s not just store bought milk that can benefit from Calcium Chloride. If you are using Goat’s Milk for cheese making, particularly when making hard cheeses, you may need to add Calcium Chloride to get a firm curd as Goat’s milk goes through a natural homogenization process in the animal’s body and without CaCI2 it may produce a curd that is too weak to cut properly."
Continuing on in that particular article – and cross-referencing with other online sources, I believe Calcium Chloride is used in the cases of
  • Store-bought pasteurized milk which needs a boost in calcium.
  • Goat milk which may have lower calcium levels than cow’s milk.
  • Raw cow’s milk which may be deficient in the mineral due to poor feed quality.
It also seems that Calcium Chloride should be added only in small amounts – 1/4 teaspoon per gallon of milk – and mixed with the milk before beginning the cheesemaking process. So maybe I could make some cheese without the calcium chloride but my feta and hard cheeses might improve because of it. Which leaves me only to try it out and see what happens next.