Ah hmmm... I have before me a PDF containing Letter 617 (the physical copy of Book IV of Selected Letters is on its way from Amazon, seemingly travelling on the back of an arthritic camel, if I can judge by how long it's taking to get here). And the family tree in the letter seems to show, first, that Cthulhu descended in direct line from Azathoth, and, second, that HPL himself is a direct descendant of Cthulhu, while Clark Ashton Smith descended from Tsathoggua's branch of the family. It is, in fact, presented as a genealogy of HPL himself, rather than anything of broader significance.
An asterisk in the tree indicates that Cthulhu was the first "of his line" to inhabit Earth.
Text of the letter is largely devoted to HPL's upcoming move to cheaper quarters.
Difficult to say from this exactly what Cthulhu's role among the GOO might have been.