Hatake clan theory - based on the literature and myths and history references in Naruto
(basically the Naruto equivalent of priests)
Kakashi’s eight dogs are a reference to the Nanso Satomi Hakkenden novel aka, the eight dog warriors.
In the story, the noble Satomi clan, is cursed to have descendants as depraved as dogs. And since the last male descendant (Daisuke) dies, the head of the clan gives his last daughter’s (Fuse hime) hand in marriage to the family dog, after the dog brings the head of the enemy.
Then the dog and Fuse go read the Lotus sutra and Fuse gets pregnant.
Then Daisuke is revealed to be alive, and comes to kill the dog (Yatsufusa) but also mortally wounds Fuse, who cuts her own stomach open to not die pregnant and the eight dog spirits come out.
Daisuke becomes a monk afterwards.
In my opinion, Sakumo is Fuse hime in this setup. And Yatsufusa could be an Inuzuka.
Fuse hime in this story is a tamayori hime archetype, who are spirit wifes, vessels for a spirit. This is going to be important later.
Kakashi means scarecrow, and acts as a scarecrow and the scarecrow god Kuebiko, being all knowing.
Scarecrows are also vessels for a god. Which is used later, when Obito uses Kakashi as a vessel for the fight against Kaguya.
Kakashi is called Kakashi of the sharingan and Sakumo is called the white fang. In my opinion, these two could be based on the same Theme. As in they took their loved one's main jutsu and were named after it. From the data books, we know that Sakumo got his fame and probably name too in the third war. Which was probably after Sakumo met his wife.
Now onto the Sage of six paths connection. Naruto is kind of a rewrite of the japanese myths in many sense. Read it, or just take my word for it for now.
The myths were basically to canonize the geneology of the nobles, so sometimes the naruto charachters are following one of the famous people down the line.
Kakashi is usually a stand in for the kids of the Kotoamatsukami. And the Scarecrow god, Kuebiko.
In the Kojiki, the one who got Amaterasu’s mirror, which is half of her soul is Ame no Koyane, who was one of the first Amaterasu priests. (Kakashi is referencing both)
(And Obito is usually a stand in for Amaterasu )
There is this one eyed samurai, who is called the one eyed dragon, Date Masamune (who is very moon themed) ,and is claiming descendansce from Ame no Koyane. So I think we can agree on the refence.
(If you don't know this man, what have you been doing with your life?)
Ame no Koyane is the son of the kotoamatsukami Kamimusubi, who brings Okuninushi (Naruto) back from the dead, and he has a kid (Ame no koyane). I would say, that during the runtime of Naruto, the kotoamatsukami were one person, because of Shisui’s Kotoamatsukami, and that one person was the sage of six path. And that’s why Hagoromo was talking about love with Kakashi. (Even if it made no sense of canon family tree of the last 20 chapters)
Kami musubi was also the one who collected the corpse of the food goddess (Kaguya. More on this later when I'm finished with the Ainu legends). This was the beginning of grains and agriculture. Also, the food goddesses eyes turned into the rice seeds, if anyone cares.
And the Hatake are always named after Agricultural things.
The children of the Kotoamatsukami that Kakashi is referencing, created one clan, the Fujiwara, who often intermarried with the royal family, aka the amatsukami, aka the Uchiha.
And since Shisui had the kotoamatsukami, Kagami might’ve been half Hatake, or somewhere down the line.