The way the “loudness wars” work is by reducing the dynamic range to be far narrower than what’s available at 16bit, which gives the impression of loudness even though the loudest sounds are no louder.

Lets say the mastering process compresses the recording extremely to a 42dB (8bit) range - outputting that to either 16 or 24 bit will make zero difference. The audio will still have a dynamic range of 42dB.