Some trans people set a standard for themselves as what it means to be their gender. As a trans woman myself, my personal ideal is rather traditionally feminine, though in my case I've come a decent ways in making exceptions for typically masculine hobbies or interests I still retain. They are not "men's hobbies/interests" or "women's hobbies/interests" any more, they are my interests. That out of the way though, my point was going to be that Chiaki's ideal for masculinity seems to be set in his father.
So, for Chiaki, presumably to be a man is to be like his father. His father is a hunter. For Chiaki to be a man, by his own standards, is to be a hunter. And above all this is to be a man and to be accepted by his father, so for his father to finally agree to hunt with him is his father seeing him as the man he wants to be.
Maybe if this were a longer series, we might see Chiaki coming to terms with some feminine interests he has and finally internalizing them not as gendered interests, but as his interests. This is a one-shot, however, and I personally think it handled the aspects that it did just fine.
I would like to hope that we can all agree though: Takatou is a man among men and sets a high bar for friends. You rock, Takatou.