That may be true for all the practical cases for the OP, but it's not generally true. A trivial example being several of the polar maps. In those cases, the equator would give terrible results, and the center pole would give similarly good results.
For anyone interested, some really interesting stuff is at http://www.progonos.com/furuti/MapProj/Normal/TOC/cartTOC.html, especially the "Mathematics of Cartography" section.
When your only tool is a hammer, all problems look like your thumb.