I don't pretend to be an expert, and apologies if much of this seems very basic/obviously, but here goes with my theories about why some photos seem flat while others give a sense of depth ...
A photo is a 2D representation of a 3D scene (obviously), so if we perceive a "depth" to a photo it's because we are picking up clues in the photo which our brains are using to interpret the scene and interpolate the sense of depth.
One clue to depth is from shadows, so directional light used well helps give a sense of depth. I think you can see that in a number of the photos you linked, including "Otto" and this one https://www.flickr.com/photos/mariazuelo/30449645921/ Even a slight vignette on the image can sometimes help give a sense a depth.
Another clue to depth is from in focus areas versus blur. That's clear enough when you see background blur, but I think it's often particularly noticeable when you have blurred foreground areas as well, eg in this photo I found on Flickr https://www.flickr.com/photos/behzad_rad/32318420484/in/pool-zeissotus55/ where you get a clear impression of depth "layers", and I think you can see it in this photo https://www.flickr.com/photos/95698098@N06/32482571716/ which you linked to earlier, where the tip of the nose is blurred.
Another clue can be the relative sizes of objects (which means choice of focal length and distance to subject are at least potentially relevant). I think the "Otto" photo is an example of that, with the guy's nose being a little on the large size. You can argue about whether or not its flattering, but I think it does help add a sense of depth to the photo. In the case of portraits, I think it also tends to make you feel like you are standing closer to the subject, which is an effect I actually like sometimes.
There are other things too of course, eg straight lines getting closer together, such as looking along a railroad track, etc, etc.
All of that said, some lenses do seem to be better than others at giving a sense of depth. The ability to blur out of focus areas is probably part of it, but I agree it's not the whole of it. My belief is, at least in the past, lenses designed for portraiture (eg 50L, 85L, etc) have tended to "suffer" from field curvature issues, ie they don't have a "flat field", and I wonder whether something about the lack of a flat field actually helps in creating a sense of depth? I think relatively recent lenses have tended to have flatter fields, so perhaps I'm on the wrong track there. I've also seen someone on CR (can't recall who), saying that higher end lenses tend to use glass elements in places where cheaper lenses use polycarbonate elements, and the glass elements do a better job of creating a depth effect and "pop".
Regarding images where the subject seems like a sticker on the background, my feeling is that tends to happen when the subject is the nearest object in the photo and the background is quite blurred but obviously not that far away, and especially if the light is "flat" as well. And I wonder whether it can be exacerbated with newer lenses (like the Arts) which are very sharp even wide open, because they can create such a precisely defined edge to the subject?
This 50 Art shot I found on Flickr https://www.flickr.com/photos/148209260@N04/32039657674/in/pool-sigma-50mm-art/ almost looks like the subject is a sticker (I think) if you look at the right side of the photo (eg subject's left shoulder). However, her right side gets blurry around the shoulder and upper arm, giving the impression the subject is attached to / part of the rest of the image, so in the end I don't think the "sticker effect" is really there. Similarly, I think you can see it starting to happen in this photo https://www.flickr.com/photos/orsonwang/32615125323/in/pool-canon35mm_is_usm/ if you just look around the head and next, but again it's counteracted by other parts of the photo. In contrast, I do think you can see the "sticker effect" at least somewhat in this shot https://www.flickr.com/photos/128693487@N08/32497215313/in/pool-zeissotus55/ and this one https://www.flickr.com/photos/ileohidalgo/32884247933/in/pool-2064616@N24/ . I have a particularly good example of the "sticker effect" in a shot I took last year with the 35 Art, but I don't have it handy to upload it.
I realise the examples above come from a number of different lenses, but they are just what I found having a quick look around flickr now.
Anyway, at least generally I would choose a photo which gives a sense of depth over a "sharper", so I'm intrigued by your belief the 50 Art does not do a good job of creating a sense of depth - although I'm not ready to say I agree that's the case
Sorry for the long post, but I find this quite an interesting topic, as you may have gathered!