WARNING: The following contains spoilers for Boruto: Naruto Next Generations Episode 193, "Coexistence," now streaming on Crunchyroll.
Naruto adopting Kawaki was one of the Boruto manga's most shocking moments as it broke so many protocols in Konoha. At the Kage summit, all the village leaders agree not to confine the Kara weapon, but no one expected Naruto to bring it -- or him --home, least of all Boruto. It felt a bit too sudden; something that could endanger the Hidden Leaf without a more formal stance.
Interestingly, as the anime now details this particular arc, it makes a big, positive change in terms of the reception to Kawaki at home -- which informs how Boruto warms up to and proceeds to build a relationship with him.
In the manga, Naruto brings Kawaki home from the Ryutan lab after learning that he keeps trying to run away. When they get home, Naruto has Hinata make them dinner, only for Kawaki to lash out when Naruto tries to probe for information, including his name. Kawaki's distrust of Naruto leads to him acting out and breaking the vase Himawari made for her mom's birthday, only for Boruto to come home and start a spat Naruto breaks up. The pair soon forge an uneasy truce for the sake of keeping the peace.
Really, the way the adoption plays out in the manga feels like it was just ticking off boxes. The anime, however, smartly course-corrects in Episode 193, creating a more nuanced version of events: Boruto meets Kawaki at home before dinner and it's obvious they don't like each other. They trade words, but after his son basically deems Kawaki an outsider, Naruto takes Boruto to his room for a chat.
He doesn't like how Boruto chided Kawaki, insulting him on something out of his control. Naruto shares a lesson from his past, telling Boruto about how he too was misjudged. He mentions how the Hidden Leaf scorned him, which could have seen him harness Kurama's powers and try to destroy Konoha (like Sasuke). He makes it clear Kawaki didn't ask to be made a tool by Kara, so they have to bear with his angst and understand why he is so aggressive and distrustful.
Naruto preaches love, compassion and empathy, which is what healed him when he was broken. This properly informs why, when Kawaki flips out at the table, Boruto decides to forgive him. It isn't random, or anything to conveniently move the plot along -- it's heeding his dad's words and understanding why his rival is in pain. This encourages Kawaki to make the first big step: sharing his name.
The moment brings Naruto's story full circle, going from someone society shunned to someone with not just the power to protect those going through what he did, but also the power to teach others to help in the same way. Teaching Boruto empathy and then seeing his son understand enough to make an effort with Kawaki is something that should give the Hokage hope.