Actually, it seems from the article here, that Sagawa spent extensive time exploring breath power, and at length determined, for himself, that it is optional. While I have no doubt that this originated as some Taoist practice in China, I wonder if this was actually an enunciated part of Takeda's system.
I think that what Takahashi was trying to say was that it is not
optional - until you reach a certain point, as Sagawa did. It makes sense, elementary school children learn certain drills - but that doesn't mean that they use those drills forever.
It's clear from what he was saying that it was an enunciated part of the system - note that it appears on the densho, and that he refers to a series of breath exercises taught systematically. Also, we know that other people who trained at Sagawa Dojo have brought back breathing exercises from Sagawa directly. And of course, there is Tokimune, who also spoke about and taught breathing exercises.