For one thing there is massive distortion of relative distances in southern parts... New Zealand or Australia to Antarctica is something like 4 hours flight, not halfway across the planet.
0. http://www.spoon-tamago.com/wp-content/uploads/2016/10/autha...
(Yeah, you can make a map where straight lines work, but only for a chosen point of origin or antipodes around that point.)
What I said is impossible is to have a map where all routes between pairs of points look sensible. It gets worse if you optimize for other factors, like making maps equal-area, conformal, rectilinear, etc.
There's an easy way to make all routes between pairs of points look sensible at once: plot them on a 3D globe ;)