Right so I’ll try to give as brief of an explanation as I can now from my point of view. Now history wise, you summarized it but missed a few key points. The Hayabusa for one are worshippers of the Great Three Dragon Spirits, which as you mentioned protected the Hayabusa during Kaguya’s wrath on the lands. The Dragons in turn due to the Hayabusa’s relentless faith in them awarded them with their powers. This being the power to control Wuji or Nothingness. Each of the three dragons represent one of three unique chakra pools or “Dantian”; these pools being what the Hayabusa use as their means to control Wuji. They are referred to as “Lower”, “Middle” and “Upper” each one representing a specific aspect. Lower is solids/tangible/liquids etc, Middle is energy i.e fire, lightning, plasma etc while Upper is Spiritual, i.e Yin, Yamanaka Hiden, Genjutsu etc. The first era of Hayabusa was “gifted” the power of Wuji much like how Hagoromo was gifting chakra to people, but as the generations pass, Hayabusa are now all naturally born with the Dantian within them. So in hindsight, it became the clan’s kekkai genkai so to speak. Now you said we are granted three different types of chakra, which is incorrect. The three dantian I mentioned before is what you meant, and with the dantian comes some inherent passives that the clan has. “Wujin”, “Taijin” and Spirituality. These are passives that all Hayabusa are born with; basically, the Dantian being passively active at work protecting you in certain ways. Now when you weaponize these dantian with your regular chakra, it mixes the unique energy released by the Dantians in order to create “Dragon Chakra”. Think of it like Senjutsu, where you mix Natural Energy and your normal chakra together, the merge into one in order to become Sage Chakra. It’s the same for Hayabusa but you don’t need to absorb anything or whatever unique features lol.
These passives have some active applications we will go over soon, but aside from this do you have any questions so far? If not then pick one of the passives you’d like to go over first and we’ll discuss it.