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.