Interestingly, and this is a hard one to swallow because the rule has been around a long time in all rope disciplines, testing has shown that treading on ropes does nothing to diminish rope strength over time and doesn't significantly introduce dirt/grit into the core fibers. Grit gets into ropes just fine without our help walking on them. But... big but (not butt) this "truth" is very similar to a parent telling their child, "Don't touch baby birds in their nest, your scent will cause the bird's parents to reject them". Very few bird species have a sense of smell and the ones that do don't nest in your backyard unless you happen to have an albatross or turkey vulture in your backyard. Point being, false rule but very effective at keeping kids from disturbing bird nests, and getting bird lice on their hands ;-)
Likewise with your rope, stepping on it means you're not paying attention, you're not respecting that which keeps your life safe and you're probably not paying attention to a bunch of other things you should be paying attention to besides your rope. False story but very useful way to help climbers respect their ropes and pay attention to their surroundings. Very useful and effective story, that's why both myths have stuck around for a long time, they produce good results.
-AJ