Aye, it took a while for ye olde brain. I was fixated on the "volume control" part, but it's really all about the harmonics, right?
In that case, a word of caution: Emulators are often misleading in that respect, actual hardware can sound quite different. Generally 48k tends to be more noisy, and harmonics tend to be buried under the noise. Anyway, if you're going to explore further in this direction, I can help with some hw recordings.
The question remains where these harmonics come from in the first place. Is it because at these marginal pulse width threshold overflow errors from the frequency calculation become more significant?
When I worked on those multi-core engines with many volume levels, I also noticed another effect. If there are several consecutive frames with high total volume (mimicing a DC offset), some sort of volume ramping will occur, gradually turning rectangular waves into saws. Octode2k16 is perhaps the best example of this. I tried to control this effect but it was very unreliable. Still wonder though if it can be used somehow.