The GP post is essentially correct. I did a bunch of the math behind it in my undergraduate thesis / research project in the context of high-speed PWMs for motor drives.
Essentially, if you reconstruct the sampled information with a non-ideal DAC converter, it phase shifts the output based on the time varying magnitudes of the input signals. When analyzed mathematically, the effect is very similar to phase modulation (PM) or frequency modulation (FM). Normally, phase and frequency modulation is used in the context of radio receivers, which use complex filters prevent distortion. The audio amplifier has none of these filters, and the result is that the phase modulation shows up as audible distortion inside the normal frequency band. The effects of this distortion are significant. I noted them in the context of a motor drive.
Modern DAC manufacturers are well aware of the fact that their products are non-ideal. As such, almost all of the new audo DACs feature circuits to reduce the distortion. However, this distortion ellimination isn't perfect, especially for a 16-bit/44.1 kHz signal. Nevertheless, numerous papers have been published on how to create a DAC converter that behaves more closely to the ideal DAC converter modelled in Nyquist sampling theory.
Realistically, the bigger problem with the 16 bit/44.1 kHz format is the loudness wars.
The loudness wars cause clipping. No amount DAC converter trickery can fix clipping. The result is that people say old LPs now sound better than new CDs. They are correct. The old LPs were mastered with more dynamic range and less clipping than modern recordings.