Well, nice analysis, but still incomplete. Where did Hagoromo get that chakra to begin with? How did he get it? The answer to these two question are the real source of Rinnegan.
Maybe on how Hagomoro received the Rinnegan, but not how Madara did.
The mixing of Ashura's chakra and Indra's awakens the Rinnegan. As of now, only the Uchiha have managed to do that. We still don't know if it is possible for the other transmigrate (i.e. Naruto or Hashirama) to have been able to do it as well.