There are three separate things going on here: The standard "6 dB per bit" is the approximate signal-to-noise ratio with a flat dither. Then you can use a sha

There are three separate things going on here: The standard "6 dB per bit" is the approximate signal-to-noise ratio with a flat dither. Then you can use a shaped dither to increase the dynamic range at some frequencies at the expense of others (the SNR actually decreases as the dither is louder overall, but quieter at frequencies the ear is most sensitive). Then in a real-world DAC there is analog noise getting in at various points in the circuit, so that is what limits the end result to 111 dB. So yes, the main reason to use a 24-bit DAC here is for volume adjustment, so the lowest bits are well below the analog noise floor, not above it like they would be for a 16-bit DAC.