The propagation of maps which (essentially) lie about the relative sizes of all countries/continents within the northern hemisphere seems to be the basis of something beyond "a convenience for sailors", it's the basis of covering up the truth.
Note that there is only a small amount of shrinkage in the countries within the southern hemisphere. And the more advanced of these are in the lowest lines of latitude.
I would conclude that maps like these are very favorable to the FE assertion.
Where these sizes come from is pretty simple, globe-based map data is re-projected from a sphere to a plane. Since you can't flatten a sphere into a plane without stretching, distorting or cutting it, you get all the various types of maps. The latitude and longitude lines stretch with the map, so are still useful in locating an absolute position. London will always be at 52° N, 0° W no matter what shape you stretch a map into.
The Mercator projection is the most common, and the one shown here. It stretches the map at the poles which is why the continents up north look so large.
The reason the southern side doesn't stretch as much is simply because there are not as much land south of the equator, so there is less to stretch. The animated map cuts off most of the southern hemisphere, easily noticed as it's missing Antarctica. You can see the full projection below.
The further away you get from the equator, north or south, the more distortion you get using a Mercator projection.
There are a lot of better flat projection methods in my opinion, but they all have same problem that you can not represent a sphere with a plane. It's just not possible to do so without distorting and all flat maps have to pick how they distort. The best way to view data plotted on a sphere, is on a sphere.