Nope. Although they are both conventions, in the electric charge there is really no natural choice. While mathematicians can argue whether or not a factor of 2 is a natural choice for a convention.

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.

I use a different definition of natural. The fact that the electrons, negatively charged, were moving wasn't known when they assigned the positive-negative labels. But still, although what you are saying is certainly true in proton/electron systems, it would be completely reversed in antiproton/positron systems, where the charge would be moving together with the particles. Also, don't forget of ions, or particles in space, or in particle physics experiments, where both charges move.

The point is that the natural choice that you assign would be natural just because of some contingent conventions, but it's not more natural in terms of some more fundamental/mathematical meaning. While the pi vs. tau is.