You might want a high bit depth audio stream if you were seriously boosting the volume, but not if you were lowering it. At most you might want a 24bit DAC.
And if you can't pass an ABX test, then you don't really want it. Can you?
Why would s higher range not be good for lowering the amplitude? If you divide the samples in a 16bit stream by e.g 16 that means losing several bits of range. Surely if a 12bit dynamic range was perceived as being just as good as 16bits then we'd have that to begin with?
To be clear, when I say "lower the volume" I mean lowering the amplitude of the digital samples, passing them through a high quality dac and amplifying. I.e sliding the volume slider down on a iPhone using AirPlay to a Toslink->amp.
And if you can't pass an ABX test, then you don't really want it. Can you?