“why is this so important that it should come out?”
Because, if you think about it, if he’s not a hunter and does not own guns, but says that he is and that he does, that makes him less than honest.
That’s an important trait.
Maybe he's a "closet hunter" and has only decided to emerge recently. Maybe he's tracked all kinds of stuffed animals lurking in his closet all these lifelong years. Come on, now, folks, let's give him the benefit of our doubts, shall we?