Why does it have to be a single continuous map at all?

Couldn't we use multiple circular projections centered at different latlongs? We'd give up continuity at the seams, but at least every region could be shown close to the center of the projection with little distortion. I would certainly prefer that in an educational setting (drawing the globe from ~6 different latitudes onto a 2d map should be sufficient).

And when drawing a map on the computer, the projection should be centered on whatever part of the globe is currently at the center of the screen.