"...What's special about adding the stearic and glycerin to the cook after the batch has turned to mashed potatoes? Why couldn't you add everything to the pot at once?..."
Adding stearic and glycerin "super cream" after the cook does seem inefficient and rather silly at first glance, but I strongly suspect there are good reasons for doing this. It's too bad the theory and practice of cream soap making seem to be shrouded in mystery, but it is what it is. Being a curious sort (as y'all have probably noticed), I did my first cream-style shaving soap a few weeks ago following Lindy's basic procedure. I evaluated the process as I made the soap. Given my background in chemical process engineering and soap making, I think I'm qualified to take a stab at answering these questions, subject to revision as I get more experience.
First off, stearic acid has two basic functions in the bath and body world -- stearic-as-soap and stearic-as-thickener. The two are not necessarily the same. Soapers don't usually think of stearic acid as a thickener and texture modifier, but lotion makers use it routinely in their products for these benefits.
Stearic acid added after the cook will help the product to be thicker and less sticky. If you added all of the stearic to the soap pot, the other fats in the recipe are much more likely to be the "superfat" since stearic saponifies so readily. This is not going to give the same results of thickening and texture modification as when you would add stearic after the cook as a "superfat" aka supercream.
One of the complaints about adding stearic as the thickener is that the cream soap can end up rather waxy feeling, so it's clear that stearic-as-thickener should be used with a light hand.
Adding glycerin after the cook will act as a processing and mixing aid. It will help loosen the sticky, heavy soap so you can more easily mix the soap, incorporate the stearic, and whip the soap to a lighter texture (if that's what you want to do). If you add all the glycerin to the soap pot, it will not function as a processing aid at the time of mixing.
A close analogy is making a cake by hand. Some recipes have you cream the sugar and butter, beat in the eggs, and then add the flour and milk, alternating about 1/4 of the flour, 1/4 of the liquid, 1/4 flour, etc. until all of the flour and milk are stirred in. You could mix in all of the flour in one go and then mix in all of the milk, or vice versa, but the process would be considerably more difficult.
One could use water as a processing aid, and I think some cream soap makers do, but water evaporates, so the end texture of the soap will change with time. Glycerin has its drawbacks too, but evaporating is not one of them.
Soap is more soluble in glycerin than in water, so that is another consideration for using at least some glycerin for efficient processing.