My method for hard boiled eggs is a tad different. I get water to a roiling boil then add eggs. Cook for 20 min. As soon as they are done, I drain and add cold water. Might do a couple of 'cold baths' and then I place some ice cubes in the water and let sit until very cool.
I lay out paper towels, crack and roll eggs on paper towel and begin to peel. If I have a stubborn spot I will continue peeling under cool running water. Once all eggs are peeled it's pretty easy clean up just wrapping egg shells in paper towel and tossing. And rarely do I have an ugly egg!