Out of curiosity, why do you want all this heavy-lifting worldbuilding? What specifically are you trying to get to?
In my experience, no one asks for this level of detail unless they have an exploit in mind or they are looking to trip the GM up on an inconsistency for the fun of it. (A cynical worldview, I grant.) What specifically are you looking to do here? Simply telling us that will allow us to tailor the focus of our answers.
Well, the Hagoromo are the closest we have to a rival clan in Leaf and I still struggle to understand their power base. I also am a fan of cultural verisimilitude and currently it still feels like Leaf believes in [placeholder religion] that is designed to be more about simulating the challenges that religion posed or poses IRL than really being adapted to the local history and lore. But religion is such a big part of a society's history that I can't quite imagine that you just blindly showed it in to Leaf, not with the massive amount of hidden cosmological world-building you have implied and hinted at throughout the quest.
As for tripping up GM inconsistency "for the fun of it", I don't actually know if that's a fair assessment. Is it "for the fun of it" if the inconsistency in question makes the whole thing less fun for me? The inconsistency being the (on the surface) apparent lack of reconciliation between the Will of Fire being an old and established religion with idiosyncrasies and baggage on the one hand and a philosophy set up and spread by an intelligent ninja that revolutionized the world's social structure ~100 years ago on the other.
Also, the Will of Fire is not just some background world-building trivia. Our plans and our IC loved ones have been impacted by it multiple times and the majority of named characters around us, be it ignorant peasant folk blindly believing something they don't understand or highly intelligent and powerful battle wizards and political leaders like Naruto and Asuma, credibly profess to base their personal morals and ideals on.
Even from a pure gameplay perspective, players were about to commit quite large blunders because they didn't understand the Will of Fire's sociological underpinnings, like when they suggested undermining it by introducing and fostering new religions in Leaf. Luckily IC advice stopped that.
Or to put it in very simple terms: The Will of Fire is being used as a cudgel against us and one of the reasons we are vulnerable to it is because knowledge about it that seems like it should be available to us IC is not available to us OOC.
And beyond that, the Will of Fire seems to me like one of the most important world building blocks of Leaf society right after a) the clans, b) the clanless situation and c) civilian economic status. It is also closely intermingled with all three of those.
So both from a gameplay perspective and from an enjoying and exploring the rich world you created perspective understanding the Will of Fire seems important to me.
But if I am asking for too much, then I guess the most actionable questions I have are these:
A) Are there purported actions or words of Hashirama or Tobirama (or Hiruzen I guess) that can be easily extrapolated into a clear homophobic message?
The answer would change how we tackle that whole social issue.
B) Is there anything known and actionable when it comes to the Hagoromo making the switch from whoever they were before Leaf to becoming the main and most respected teachers of Senju Hashirama's philosophy (and/or its cosmological aspects) that would allow us some advantage if exploited?
The answer might allow us to undermine the Hagoromo in the longer term by letting someone else take their place and/or by relegating them to being the foremost scholars on pre-Hashirama spirit and cosmology knowledge only instead of them having such a stranglehold on Leaf political philosophy as well.