SMSL has some DACs that allow for Sound Color, and there are options for emulating actually scientificaly ideal tube amps, creating only perfect 2nd harmonics, no bad 3rd harmonics.
So if you wanted to, this option could beat a +100k tube amp easily, made possible by one single neat chip. But for me this sound color option, even if scientifically perfect, causes faster hearing fatigue.
3rd harmonics have their use. If you want more 'analog roundness', then yes, 2nd sounds better. If you want to excite something, such as the high end (cymbals) giving it more "spark" and "sizzle" or liven up a muffled choir, then it sounds better with both 2nd and 3rd harmonics.
Your post is I think a prime example for an idea I wanted to communicate: any produced song already has a very large amount of saturation on it, any more would be too much.
As an example, "Clean recorded vocals" have been going through the staple LA-2A leveling compressor right at tracking (before a mixing engineer get his hands on it). Measurements here suggest that adds around 3-4%THD during the rising stage (so going from uncompressed to compressed state)
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Then when the mixing engineer gets his hands in it, they'll add a little saturation here, a little extra compression there. (Compression by definition adds saturation during the adjusting of levels, so during the attack and release movement) .
One of my compressors shows THD, and I'm not keeping track but if I had to guess I think clean tracks get about 0,1% at RMS levels and maybe an additional 1% at peaks. When we are talking about a vocal with a distinct modified sound, The Roots - The Seed 2.0 for example, it's at least double. The drums in that track sound to me like 5-8% THD from compression. In fact, that whole track sound super crunchy. They key sound of "Lonely Boy" by the White Keys is saturation, there's nothing on there remotely clean. While the drums sound relatively clean compared to the vocals, if the drums were actually clean, they would stand out like a sore thumb.
Cleaner genres like classical will have a lot less of this, but still some to aid the hearing of the listener (traceability of individual instruments). Edit: classical is often pretty compressed as the enormous dynamic range in auditoriums just doesn't make sense for living rooms.
The mastering engineer will then add little bits of saturation here and there to balance it all out, and will of course limit the track to get the loudness up to genre standard. Although mastering compressor/limiters, such as the Weiss DS3, are extremely transparent (they hide the saturation in auditory masked frequencies).
Point is, it makes sense that you find it tiring. The optimal amount of saturation has already painstakingly determined, any more is going to tip it over the edge. And that is for well engineered songs. Fast pop is overly crunchy, as it overly saturated gets more attention in the short run compared to clean or under saturated, and is thus even less accepting of even more distortion.