On the other hand, humans care a lot about climate change and this affects the poles a great deal, and maybe a projection that centers the poles like this can help make these regions of the world more visible.
I happen to originate from a different country than where I currently reside and on this map, the path that the plane takes makes a whole lot more sense. It seems to be a much better model of the world, to get better, you'd need a 3D globe.
However, I don't want to go too hard on these guys. Reading the arXiv paper, I think it's clear that they wanted to make a cool set of disks for the Solar System. They actually look really cool for Jupiter and Saturn with their equatorial cloud bands and polar vortices! And centering on the poles gives a nice symmetry with the Earth and Mars with the ice caps.
They also show the version of Earth with the projection centered on the Western and Eastern hemispheres and the distortions are a lot more obvious (e.g., Alaska, Canada, and Australia are pretty bad). I can see why they like the North/South pole version.
This feature of the map seemed particularly unique:
"To measure distances from one side to the other, you can use string or measuring tape reaching from one side of the disk to the other, he suggested."