That was the point of my reply. Of course Chi Omega was never a spin-off of anything. I don't think that idea had been suggested.
I have been a Chi Omega for over 44 years now (Mu Chapter, University of California, Berkeley), and have worked as an alumna for Chi Omega ever since, wherever I have lived (when in the USA). As I have written extensively, there is a difference between present-day legalities and historical lore.
At the very least (and possibly because of Dr. Charles Richardson's fraternal affiliation), Chi Omega's historical "connection" with Kappa Sigma is undeniable. It is celebrated still today on many campuses, and both fraternities are the better for it.