I mean think about it Hagoromo didn’t need to gain someone dna or other eyes, yet he has Rinnegan
[Kaguya] Yang = Byakugan/Tenseigan and Chakra Mode - weapon or cloak
[Jigen] Yin = Sharingan/Rinnegan and Sage Mode - tattoo
Yin has the chakra residing in the brain which is reflected in their eyes.
Yang has the chakra pooled in the heart and is reflected by being engulfed by either tattoos or chakra.
Kaguya [Yin-Yang] gave her a mature doujutsu on top of her fully matured "chakra" Mode.
*Please note: we never seen these abilities because Kishi has not thought of them OR was hiding the events in The Last and Boruto The Movie, so we only see the base So6P abilities that we saw from Obito, not even Madara's level.
So this is merely speculation but it is heavily hinted that this is the case as this is what we had seen with Momoshiki and Kinshiki. As well as Toneri's requirements to his So6P (Yang edition).
Kaguya, as well as Momoshiki, were the only 2 that we know who had attained full convergence of Yin and Yang. Kaguya being Yang first and Momoshiki being Yin first.
However, for Hagoromo as well as Hamura, they did not need to have these requirements as their DNA is pure AF.
They did not gain the full Yin-Yang abilities from the start.
Hagoromo eventually did gain SM (his mark on his forehead, similar with Hashirama's forehead Sage Mode, which is further backed by Tsunade's Yin Seal on her forehead.
SO NO, IT IS NOT HAGOROMO'S RINNE SHARINGAN THAT IS SEALED)
Hamura also gained his Tenseigan eventually as shown in The Last.
These matured on their own as the powers that they gained from their mother is within their affinity instead of being opposite of their affinity.
10tails being chakra (Yin) went to Hagoromo (Yin).
Gedo Statue being vessel (Yang) went to Hamura (Yang).
Knowing the affinities' characteristics, Hagoromo gained his Rinnegan and SM from the very start while Hamura gained them [Tenseigan and CM] at a later time.
The thing is he has done something that no dojutsu has ever done which is learn Sage Mode. Indra, Itachi, Obito, Madara, Uchiha clan, etc, hasn’t tried to learn Sage Mode. No Hyuga or Otsutsuki has learned Sage Mode.
Just imagine a clan full of Rinnegan, that would be overpowered. They have access to their Sharingan/Rinnengan powers.
Madara and Obito did not need to learn SM due to Hashirama's cell. They met the requirements of having SM and doujutsu.
"Why didn't Sasuke naturally gain Rinnegan?" His SM was from a curse seal. I doubt he really trained Sage Mode. If his is legitemate, then IDK. Perhaps it is not strong enough?
"Why didn't Obito naturally gain Rinnegan?" Because his MS is not EMS, unlike Madara.
Hyuga with Sage Mode would do nothing but boost their physical traits.
Remember, Hyuga is Hamura's route. His route consists of Chakra Mode and Byakugan. So gaining Tenseigan wouldn't work if a Hyuga practiced Sage Mode.
Also we do not know whether or not THE Sage Mode can be taught or whether it is inherited. If we use the Hamura route, we can assess that the Chakra Mode can be inherited as seen with the Uzumakis.
Perhaps that is one of the reasons why Sasuke could not naturally get his Rinnegan. It could also be based on maturity. It is possible that it might take some time for it to mature.
"But how come Toneri's Tenseigan matured in mere days?"
DUDE, IDK? Bad storytelling is consistent with Kishi. Just like how Kaguya was lame and didn't exhibit new abilities. Just teleported everyone in different realms.
Anyways, it seems the theme is consistent with Part 2: Yin and Yang. Yin being represented by the Mind and Yang, the Body.
Yin = skill through talent, Yang = skill through practice.
Yin = fast maturity, peak performance, Yang = slow maturity, almost no limit.
Symbolisms:
Mind = Kakashi, Body = Might Guy
Mind = 10tails, Body = Gedo Statue.
Mind = Rinnegan/Tenseigan, Body = Sage Mode/Chakra Mode
Mind = Susanoo, Body = Avatar
Senju' seals are found in the forehead or the brain.
Naruto's seal is found on his stomach or the body.
8th gate is on the heart
RinneSharingan is on the forehead.
I believe that in order to unlock doujutsu or mode, you have to get it from a strong lineage.
Which begs to question: Did Hashirama actually get the God Tree Sage Mode from his lineage or was it taught to him by chance?