The tradition of breathing with the dantien/tanden (丹田) goes back a long way in China, and is also shared with Taoism. Hakuin notes that this method was very useful for maintaining his stamina, and although is method for practice was, he said, derived from a Taoist, it'd be hard to not call his practice Zen. That said, you're correct too, that though I think Soto-shu actually emphasizes 丹田 less than Rinzai, again, due to Hakuin's profound influence. And it's not "realizing the belly," but rather "breathing through your feet;" in order to do that, as Hakuin says, you have to keep the heat "flowing down." But I think all of this is simply a differing matter of emphasis.

BTW, I'm glad you're writing about sutras and what-not. And skillful means are, you know, useful when skillfully used.