Edit: For a less-snobbish presentation, try watching his Digital Show & Tell video https://www.xiph.org/video/
The basic issue with 16-bit is that low level details like reverb tails and hall ambiences or very quiet musical passages get the equivalent of 14-bit (f) 12-bit (mf) to 8-bit (ppp) sampling.
This sounds noticeably grainy and digital.
It's not about total dynamic range or sine waves, it's about the fact that human ears can do really neat source separation tricks. We can hear quiet elements in a mix without too much difficulty.
If those elements are sampled at less than 16 bits - which they will be, if the maximum resolution is only 16-bits - we can hear that too.
So 24-bits gives you effortlessly smooth sound for quiet passages and quiet details. 16-bits doesn't. (Dither helps a lot, but it only takes you so far.)
Why are there still people who pretend this isn't relevant? It's not a difficult point to understand, and it shouldn't be controversial.
Edit - the technical misunderstanding is a lack of appreciation of the different properties of the absolute theoretical noise floor of a converter, and the fact that quantisation noise isn't like analog noise. It's actually more of a hyper-objectionable and nasty sort-of-nonharmonic distortion.
So as the bit resolution goes down, the sound doesn't just get buried in noise it also gets more and more obviously distorted.
He's argued that people have done this test over and over, and nobody can ever tell the difference.
There's been exactly one serious sort-of peer-reviewed paper in the AES journal, and that paper compared high-res commercially mastered audio sources of possibly questionable parentage with a 44.1/16 downconversion.
It also included SACD, which isn't a fixed bit depth linear PCM technology, and has been justifiably criticised for it.
I'm not aware of any tests that compare raw high-res unprocessed recordings with downsampled content.
Secondly, a fair comparison would be 48/16 and 48/24.
Personally I'm not very sold on high sample rates. I know there are technical reasons why it's easier to make antialiasing filters sound transparent at 96k than it is at 44.1k, and in practice it's not easy to pull apart practical design from theoretical limits. (Nyquist is only ever an ideal. No hardware is ever Nyquist-perfect.)
Basically psychoacoustics is hard. Ears are ridiculously sensitive, brains are occasionally delusional, and marketing people lurk everywhere.
It's extremely difficult to pull apart fact from reality.
But that's no excuse for having a misleadingly superficial understanding of the theory - which the original article does.
What you need to do is get those files, and send people down to a professional mix studio. Then AB them in there. Get people to sit in the sweet spot. Then you'll have a decent result.
Thing is, people have done this very test in professional listening spaces. And the results are always fascinating (and some people can tell!). In fact it's one of the most fun aspects of a professional facility! Audio shootouts!
In fact, because the points are logarithmically spaced, the points in the quiet part of the spectrum are closer together and have better resolution than loud sounds.
There are log sampling systems - e.g. A-law and u-law are used in telephone codecs.
But uncompressed audio formats like PCM WAV are 100% linear. (Or should be, near enough.)