Great for showing the relative sizes of continents (and also visualizing how all the continents are nearly one contiguous landmass), at the cost of chopping up the oceans.
Second favorite is the Peirce quincuncial map: https://en.wikipedia.org/wiki/Peirce_quincuncial_projection , which tiles the plane.
Also it is interesting that it is as if the whole "land" of the world is basically a big island in the middle of the ocean.
I keep wondering, if there are a map like Mercator projection, as reference since everyone is familiar with it, but provide size that is relatively accurate but not 100% area accurate like Gall-Peters projection.
i.e It is a map that is not accurate by an metrics or projection in any shape or form but a better Mercator ( Normal ) Map with slightly more accurate showing of relatively size.
It is only used to provide some sort of better mental model without using the Globe. Rather than Direction And Traveling ( Mercator ), true Size, ( Gall-Peters ).
I also think that the Robinson is probably the best candidate for an accurate "default" map of the world. There are projections I prefer but the Robinson retains the recognizablity as being obviously a map of the world.
One commonality all of these projections have is that they are pseudocylindrical instead of cylindrical: lines of longitude are not depicted as straight lines but as curved lines, with the curve being more pronounced the further from the central meridian you are.
It's unclear whether they were simply made aware of different projections through the show or if they thought Gall-Peters was the ultimate solution, as you assumed.
The Gall-Peters projection is just as problematic as the Mercator projection. Actually, it is arguably more problematic because it doesn't have any compelling features that it does better than other world-scale maps.
The best world-scale maps are going to be along the lines of the Robinson projection, where you give up on trying to pretend that the world atlas is a rectangle. You're going to be screwed any way you go about it. Cylindrical map projections suck at large scales, because one of the directions is being stretched way too much. Conic map projections are good at intermediate scales (state-to-continent size), but when you expand the map to large sizes, the distortions at the edges become too great to bear.
The Gall-Peters projection is an equal-area map, but it goes about this in an especially annoying way. The areas near the equator are made excessively tall and skinny, and near the poles, they are excessively short and wide. And it turns out that human beings are really not good at comparing sizes of rectangles that have drastic aspect ratios in different orientations. The spaghettification of countries doesn't lend itself to a visually pleasing map, and it really obfuscates useful tidbits such as "Maine is closer to Africa than any other US state", which is pretty clear in virtually every other world map. To top it off, it was presented to a cartographic conference as the map that's better than Mercator at a time when all cartographers were already using non-Mercator maps and complaining about lay people relying on Mercator maps for world projections.