Agreed. There is no sane reason why live translation and/or its privacy properties should depend on the specific headphones used. Even if the live translation were to happen in the headphones themselves, that should only tie the availability of the feature to the headphones. The privacy implications ought to be orthogonal.

I see three possibilities. Either the whole thing is made up entirely by Apple for bad faith reasons. Or some non-technical person with bad faith motivations at Apple suffered from some internal misunderstanding. Or somebody at Apple made some incredibly bad technical decisions.

Basically, there's no way that this isn't a screw up by somebody at Apple in some form. We just can't say which it is without additional information.