Would our dog get a significant nutritional benefit from our adding a raw egg to his daily meal of dog kibble? Is it quality protein and essential fatty acids he’s been missing?

You get an A+ for this insight, reasoning it out from your understanding of comparative physiology and nutritional biochemistry. As a carnivore, he is not highly attuned to and dependent on numerous, complex, and perhaps exotic, phytonutrients that might even be restricted to only certain geographic areas so as to represent a challenge if there is a mismatch in wanting to have a dog living in a certain area of the world representing environmental challenges, nutritionally. His needs are pretty much confined to what a meat eater can access. Dogs are opportunistic and will, in fact, eat eggs when hungry, and in nature they have access to duck eggs, turtle eggs, and so on, as well as some insect eggs and larvae. As you know as well, these are all sources of protein and fatty acids. So we think your idea is an ideal way to supply your dog with what he is needing now, and this should satisfy his system. As you are wondering, several times a week would be sufficient here. That need not be done every day.