Professionals on the fire line tend to use angle grinders. By comparison, a slack belt is pretty darn tame. Most, with some experience with a belt grinder won't ruin an ax head. However, I cannot dispute the ZEN of hand sharpening.
When I worked fires, local teens from the nearby towns were hired by logistics to sharpen tools at night while the hand crews ate and slept. You could hear the grinders going for hours. I doubt many of those kids were sharpening pros. Always a good idea to check the edge on a used tool, especially those marked FSS.