The Mercator Projection distorts the map of rhe world, but what is the alternative?

I know that the way maps of the world are usually shown exaggerates the area of countries further away from the equator. 

The map of the world as we know it is based on the Mercator Projection, a way of showing our three-dimensional spherical earth in two dimensions within a rectangle devised by the Flemish cartographer Gerardus Mercator way back in 1569. It shows lines of longitude and latitude as straight lines and neatly lines up with the cardinal directions, making it easier for ships to visualize their routes along straight lines. It is familiar and convenient.

BUT ... it really distorts the shapes and sizes of areas futher from the equator, and the further away they are the more it distorts them. Greenland, for example, looks to be about the same size as the whole of Africa, whereas in fact it is 14 times snaller; Ellesmere Island, in the north of Canada, appears about the same size as Australia, whereas in fact Australia is 39 times bigger; Madagascar and Great Britain look to be about the same size, but Madagascar is actually over twice as large; etc.

Mercator Projection

This has long been known, but attempts to address the problem have introduced their own issues. 

The Goode Homolosyne Projection was developed in 1923 by John Paul Goode is an interrupted pseudocylindrical equal-area composite map, often called the "orange peel map" because it looks like a flattened orange peel. The main drawback is the "interruptions" or discontinuities it results in (which are designed to mainly fall in the oceans), although it does a good job of maintaining relative sizes of countries and regions.

Goode Homolosyne Projection

In 1954, Buckminster Fuller came up with his Dymaxion map, which maps the globe onto a 20-faced icosahedron, which can then be flattened out into two dimensions. The result is a rather strange looking shape, heavily interrupted and difficult to use, but with a reasonably accurate representation of the areas and shapes of the countries and continents, even if their alignment and positionings leave something to be desired.

Fuller Dymaxion Projection

In 1999, the Authagraph Projection was designed by Japanese architect Hajime Narukawa. Building on Fuller's idea, Narukawa mapped the globe onto 96 triangles, which he then transferred onto a tetrahedron (while maintaining area proportions) which can be expanded into a 2-dimensional rectangle. Many people believe this to be the best projection available, combining good relative  area representation with ease of use, even if the layout and directions are no longer so intuitive. It can even be folded back into an approximation of a sphere, earning it the nickname of the "origami map".

Authagraph Projection

Do you think we could ever get used to this?