This is a theory I heard of recently.
As CoLA has explained, the word "yakumo" originated from the phrase "yakumo tatsu". CoLA traced it to Susanoo's poem about founding the country of Izumo. The Izumo region actually had its own mythology recorded in Izumo Fudoki, that differs from the official, mainstream version in Kojiki and Nihon Shoki. In this mythology, the founding of Izumo also started with the phrase "yakumo tatsu".
Now, the country of Izumo is said to be a marvelous country of gods, ruled by Ookuninushi. Later, Amaterasu would grant the three Imperial Regalia to her grandson Ninigi, and send him to earth as ruler of Japan, and Ookuninushi would surrender Izumo to Ninigi - peacefully, in the official account of the story. (All these, of course, are reflections of the early history of Japan, the exact meaning of which scholars are still debating about.)
So... isn't the connection blindingly obvious? Yukari is connected to Izumo, the paradise that was. The fall of Izumo and the imprisonment of Ookunishinushi is still on her mind, and Gensokyo is a revival or continuation of the ideal of Izumo. This is what she has been fighting for.
As evidence of Yukari's dislike for Amatsukami (heavenly gods), there was this line from Subterranean Animism EX:
(Is it okay to call gods idiots?
Well, I guess it is.)
I think this theory is the best kind of theory: simple and resonant. It's also consistent with my interpretation of Touhou. Thematically, my main problem is that it introduces the idea of "good ruler" into Touhou: "the world will be right again if the good ruler is in charge", not to mention this good ruler Ookuninushi happens to be male, in what should be a story of powerful women governing themselves. I'd prefer to view Ookuninushi and Izumo as representing the ideal that formed the foundation of Gensokyo, rather than a rightful regime to be restored.
From my reading, I gathered that at least part of Japanese fandom was enraged when SSiB
Ch 19 spent nine pages talking about "femtofiber" rather than advancing the plot. They called it "proof that the plot is out of control". Whatever the significance of Ch 19 is, these fans completely missed its point.