The egg soaps I have made all have lots of bubbles and feel quite nice to use. I have also found that if there is Egg in soap made with a bit more CO than my skin normally likes, the egg (probably the protein in the egg) offsets the harshness of the CO.
So, it lends to bubbles and a very nice lather in soaps with less CO, and offsets some harshness of soaps with a higher amount of CO.
Color is another thing that egg yolks bring to the soap. Egg whites lend no color to the soap, so when I want that extra boost but no color, I have used egg white only and still had the extra bubbles & lather with no added color.
But for the best combination, in my experience, the soap with whole egg added was best.
I have not personally used the powdered versions in soap, but I did a bit of hiking in my youth; dehydrated food provided sufficient nutrients to maintain energy for lengthy trips, so I have no doubt it will do the same for soap.