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.

The mastering would have to be basically non-existent for it to use a dynamic range larger than available in 16bit, and would mean noone would ever hear the quieter parts.

If we're talking unmastered recordings for archival, yes 24bit is very useful. For final mastered copies released for listening it's completely untrue that 16bit doesn't offer enough dynamic range.