I believe it's moreso that since she has pretty much given birth to all ninja that she can use any KG as it all originated from her blood.
I mean that would make every KG a 'great' kekkei genkai >.>
Let's not get hasty, having given birth to all ninjas is a pretty far claim. She had two children, who each had two children, who went down the lineages of Senju/Uzumaki, Uchiha, Hyuga, and Kaguya.
It was retrospecting (and un-retconning) on the earlier statement/comment that the Hyugas may have given birth to the Uchiha. Since the sharingan did not show up for two generations after the head Kaguya (SoSP had the Rinnegan) and Kaguya herself seems to possess the Byakugan in both eyes, the myth that the Hyugans gave rise to the Uchiha is simply from the lack of knowledge that the Rinnegan is the Uchiha in the Kaguya Mother.
So, it would be a safe bet that, as the Hyuga are to the Uchiha, the Kaguya are to the Senju. And that all the other clans kind of just showed up once they learned chakra.