And that’s the tradeoff of using a globe, whereas the tradeoff of the Peters projection is that the map is distorted in various ways, and the tradeoff of Mercator is that the map is distorted in other ways.
Which is why that West Wing clip is, frankly, pretentious, because it doesn’t discuss the negatives of Peters, wrongly implies that it’s “the” correct way to view the world, and makes no mention of other major projections, or even a globe.
Choice of map projection is all about context. Peters is sensible in a particular context; Mercator in another; a globe in another. My usual projection of choice is Albers, which is typically great for the continental US, but I probably wouldn’t use it so much if I regularly made maps of other places!