Personally I see zero differences in the images on that page and unless the author has some really super-human vision abilities (possible! but unlikely) my guess is he doesn't either. WebP looks perfectly fine to me.
It's the same kind of artifact that makes certain movies look terrible over netflix, those that have large dark blank spaces. Maybe you shouldn't look to closely because once you see it, it'll ruin your enjoyment of certain compressed media forever.
And by the way I don't think the comparison with audiophile equipment is fair. In the audiophile case we are talking about using very similar output hardware to output what is effectively the same signal. Here we have huge differences in file size (35% and more between JPEG and WEBP, a lot more than that for true lossless), and taking diffs between them shows very much that the signal isn't the same.
There is a compression limit under which you can see it's compressed, right?
So it makes sense that there is some threshold sensitivity where a picture starts appearing "lossless". That threshold is going to be different from device to device and person to person.
I can see why it would seem like that if you aren't seeing it, but it's not the case. The differences in color banding are pretty big if you are on a screen where you can see the background shading clearly.
The brightness of your monitor and the relative brightness of your room will matter a lot. In a bright room, you might not be able to see the subtle banding in the background of the images. But if you are looking at a bright monitor in a dark room, the difference is very obvious.
You are right. I just made my room dark to try this out, and now I can see the banding!
The artefacts are visible mostly in the background, where frankly I do not care.