If I understand correctly, Mercator is the only conformal projection where north is always up, no matter where you zoom in. That makes it nice for online maps. If you relax the "north is up" requirement, there's tons of conformal projections that look nice: http://www.progonos.com/furuti/MapProj/Normal/ProjConf/projC...
I like the "classic Guyou" projection you linked to as well.
But referring back to the OP, notice the grid of books and curios along the entire neighboring wall, and the grid-like pattern of the wood flooring underneath. The Mercator projection seems of a piece with this grid theme in the room.
According to your linked image, one advantage of classic Guyou projection to the writer of the article is that it seems to contain a straight, uninterrupted vertical path through only ocean. He stated that he wanted maximize ocean in his panel cut, but I think he still had to cut through a bit of Africa.
"classic Guyou"... the sizes and shapes of the landmasses are surprisingly realistic
I don't think so at all. Australia is fairly balanced east/west in real life, but on that projection it looks like it's got an eastern tumour. Africa looks like it's got a western tumour. Spain looks like it's the size of the Arabian Peninsula. The British Isles (300k sq km) are the same size as India (3M sq km). It looks horribly distorted compared to a map on a globe.
