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It is interesting Tobirama is repeatedly used for these roles (@ Edo Tensei, @ shinobi rules). Why was Hashirama mad with Tobirama those two times? He believes in Madara and what Madara thinks about how peace will happen.
Below are edits from Naruto manga for the theory.
If this was confusing, here's some help,
Madara thinks if people wanted to stop killing each other, nothing can be hidden about their true feelings. What happens if someone is seething deep inside?
We get the answer to that.
The Uchiha died because someone was seething and hiding their true feelings and emotions. This is what angers Hashirama.
Tobirama shows both times he goes against this image of peace and endangers the other side when peace is/was attempted.
Tobirama thinks killing Madara is the way to go about it.
Hashirama wants to remind Madara about their childhood and what Madara believed in.
Hashirama tells Sasuke about how he won't forgive anyone who hurts the village. He doesn't attack Sasuke or Orochimaru. He redirects his anger at Tobirama...
because Tobirama wants to harm Sasuke. Intent or not, Hashirama stops that.
We know Hashirama doesn't believe in Butsuma who wants the Uchiha to be gone. Hashirama cares a lot about the children and he is against the idea.
Are the Uchiha gone in the present? Yes. Excluding for Sasuke, Obito, and Madara.
What does Madara say to Hashirama? The Uchiha will be slowly expunged after Tobirama becomes the Second. What does this sound like?
Butsuma, the clan leader of the Senju, thinks every last foe should be vanished and Tobirama, second leader of the village, thinks their deaths was okay because it was for Konoha. Itachi killed the Uchiha on Danzo's orders and Danzo has connections to Tobirama. Danzo picked up on Tobirama's teachings.
After Tobirama was asked if he hates the Uchiha, he claims he treats any individual who is a threat with caution and what clan the individual came from doesn't matter.
He contradicts himself in the same breath. The clan does matter.
It matters to him if it is from the Uchiha. He discriminates the Uchiha based on their clan and answers the Uchiha are threats. He has a dissonance of values because it isn't about an individual who might be a threat, it is an entire clan who are threats.
What about the children? The innocent lives in the Uchiha who aren't shinobi and have nothing to do with the Senju and Uchiha war?
If the words are slightly off, no other clans are known to have suffer from Tobirama's policies and paranoia because it has to do with Tobirama generalizing the Uchiha and retroactively seeing them as a threat (because the Uchiha did choose to stay in Konoha over Madara... This fact does nothing because to Tobirama, the Uchiha are threats if they don't prove themselves to him personally. Kagami is Tobirama's subordinate, Itachi killed the Uchiha for Konoha, and Sasuke later believed in Itachi's choice to protect Konoha).
Hashirama cares a lot about children and he was upset about how children were sacrificed in wars against Uchiha. He is the only brother shown arguing with Butsuma.
Hashirama calls Sasuke an innocent Uchiha child, someone who's got nothing to do with the blood feud, and Tobirama rejects it claiming the village is more important. Tobirama doesn't care. All Uchiha are threats until proven otherwise.
Hashirama doesn't like how this world works. People die, people grieve, people kill... where does retaliation start and end? It's so hard to keep track of the grudges to know where it's coming from.
In the end, the grudges wiped out the Uchiha.
Someone is getting coal for Christmas and Hashirama is not a Happy Santa Claus.
Tobirama's allegiance has only swapped from the Senju to the village.
It's why Tobirama doesn't care about the Uchiha massacre. What about the children? Nope. Not a single word about how the innocents were killed. Butsuma says it after all.
"Kill every last foe to pave the way to a world free of battle." (paraphrased)
That's how you get a grudge. The Cycle of Hatred.