Isn't the “natural” choice to have the direction moving charged particles to be the same as a moving charge? In everyday life it is electrons, not protons that move. When we say "charge is moving from high voltage to low voltage" we are actually saying "electron particles are moving from a position of low voltage to high voltage"--the exact opposite. How is that natural?

Even if that weren't the case, it's still a matter of opinion. Minus signs show up in a number of pretty unnatural positions as a result of the negative electric charge. The convention in Physics is that minus signs convey semantic information (reversal in direction, slowing down, etc.). The negative electric charge upsets this convention, resulting in un-semantic minus signs.