I have no idea what would happen. I guess the recipe leaves them out because they have a lot more fat than protein. And I'm not much of a baker either to be honest. Just a biologist who knows how to take care for yeast and asked her mom about the secrets of bread-making.
Maybe I could make an omelette, you're right, but wouldn't an all-yolk omelette be kind of weird? I don't know... maybe I'll do it next time :)