Lord Saito's killing off Atsu's family is the sole driving force behind the Ghost of Yotei's protagonist's quest for revenge. Sixteen years after the fateful night, Atsu is back in Ezo to kill the Yotei Six for what they did to her parents and her brother. The game begins with Atsu painting the six targets' names on a sash and wearing it across her waist.
We get to know that Master Kengo (Atsu's father) has betrayed Saito, and the latter wishes to make an example out of the former's family to deter such incidents further. Even though we get plenty of flashbacks showcasing different parts of that night and how Saito came upon their home, we do not learn the whole reason behind his actions.
The answer first appears in a Lord Saito Missive in Ghost of Yotei and then later in a conversation with the Spider. Beware, there are major story spoilers ahead.
Also Read: Ghost of Yotei review
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Ghost of Yotei: Explaining Lord Saito's reason for killing Kengo and his family

In the Lord Saito Missive Saito on Betrayal, Saito mentions:
"Master Kengo was an unmatched swordsmith, but more than that, he was my friend. Kengo was the first to welcome me home after years of being a hostage. He forged the very sword I am using to defend Ezo. So when he deserted me as my wife and children were starving... and convinced so many others to leave... I took no pleasure in the Night of the Burning Tree, but I don't lose sleep over it."
While reading this, one must keep in mind that this is merely Saito's perspective on the matter. Kengo was driven to leave because of Saito's own actions. But that is not to say we should dismiss the entirety of it either. Saito was significantly impacted by Kengo's actions with respect to himself and his family. A later section in the endgame.
After the Dragon's death in Ghost of Yotei, Saito captures Jubei and Oyuki. Atsu goes in search of the Spider to learn about where the prisoners are. She finds him honoring his brother's, the Dragon's, grave at a location where the Saito family buries their dead. During the conversation, Atsu points at a grave and asks the Spider, who mentions that it was their sister.
The Spider continues to state that their sister died from starvation after Atsu's father betrayed Saito and left. Atsu is taken aback by this information and ends up apologising.
In Ghost of Yotei, Lord Saito's revenge against Atsu's family is thus driven by something both micro and macro. He suffers personally because of Kengo's betrayal, losing his daughter in the process. He is also unable to accept the affront and what it cost him. What happened during the Night of the Burning Tree is thus done in such a way that no one else would want to repeat Kengo's actions against Saito.
Does this justify what Lord Saito did to Atsu's family? Absolutely not. But the accounts are not entirely black and white.
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