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I liked the strategy the Obra Dinn developer ended up choosing, that Plotcaster paraphrases as:

Pope’s devlog describes trying an 8×8 Bayer matrix first, then a 128×128 blue-noise field, and keeping both: Bayer for characters and objects, where the ordered grid gives a clean, deliberate look, and blue noise for environments,

My reasoning is not so much about the aesthetics of Bayer, but its tendency to create focal points that demand your attention, and cause your eye to involuntarily be drawn to them. The opening gameplay scene in Sassa'Shoom demonstrates both: the Bayer grid towards the top drew my eyes towards the sky and away from the underwater scenery, and the shading on the rocks here is "blue noise-y" in that it doesn't draw your eyes too strongly. 

Ideally, I think the focal points should be in the gameplay area, with strongest emphasis on dynamic things that you interact with (collectibles, hazards, etc), and with less emphasis for static things like boundaries - so that you can perceive their gameplay implications with minimal cognitive tax.

Actually, I thought Bayer looked really good here on the columns. It seems contradictory but maybe because the Bayer grid aligns with the columns, it doesn't introduce a new focal point.

I learned about blue noise because it's used to fix color banding in high res gradients, precisely because it doesn't create focal points. Blue noise seems to be best at this, and Interleaved Gradient Noise is second best but is cheaper on the GPU. (Examples of IGN here)

But I also agree with you that dithering can be really noisy, especially at Game Boy resolution, and it's honestly impressive how much PureKnix was able to use it and still have it be pleasant and beautiful. It seems to work best when used with adjacent tones in your palette, and maybe that's why the sky and ocean surface were more distracting than the columns.