Woody's the chill, fun older brother who also happens to be really responsible too. In other words, the perfect sub for Papa when Papa's gone at Fort Lincoln.
Unlike his father, Woody's not too big for his britches. He doesn't mind living with all the other Japanese-American fishermen on Terminal Island, whereas his father likes being "the only Japanese family in the neighborhood" in Santa Monica since he "[doesn't] want to be labeled or grouped by anyone" (1.2.2).
Woody's humble. He takes a job packing celery next to his mother (1.2.15), and he becomes a carpenter in camp (1.5.12)—this is a guy who isn't afraid to work.
Plus he jokes around with his younger brothers and sisters, like when he gets his brothers to help cover up cracks in the barracks by imitating Papa:
He put his hands on his hips and gave Kiyo a sternly comic look, squinting at him through one eye the way Papa would when he was asserting his authority. Woody mimicked Papa's voice: "And I can tell the difference. So be careful." (1.3.15)
The effect? Laughter from his brothers, which is no small feat when you consider the fact that they've all just been interned in the desert.
But he's also different from Papa in another major way: he's willing and ready to go to war for the U.S. against Japan. Why? Because as he tells Papa, "'I am an American citizen. America is at war'" (2.11.11), and because "'The more of us who go into the army, the sooner the war will be over, the sooner you and Mama will be out of here'" (2.11.13). If you're swooning right now, we understand—Woody's as good a guy as Papa is bad, which means Woody's pretty awesome.
Which is why he totally deserves his own chapter, in his own voice. But even "Ka-ke, Near Hiroshima: April 1946" isn't about Woody so much as it's about Papa's family and heritage. Consider it this way: even in his own chapter, Woody's still the perfect son—thinking and imagining about Papa and their relatives rather than just himself.
When his Great-Aunt Toyo leaves his room to let him sleep for the night, he imagines what he'll do the next day:
Tomorrow we will talk. She likes to recall those days. She says there is still a hill outside of town that Papa used to climb. Tomorrow I will climb it and see what his eyes used to see. (2.18.28)
Now can you see why we love this guy? He's all heart.