For peeling I always make an initial crack by tapping the eg on my worktop surface. Then I kind of roll it on the surface with pressure from above, sorta like the way you test a ping pong ball.
Then I make my initial shell removal at the biggest, longest crack.
There’s a thin membrane between the shell and the egg itself. If you can get under this, then the shell seems to come away in large, easy pieces. Quick rinse under the cold tap to get rid of hangers-on.
My mum used to add a dash of vinegar to the water and my sister strikes a coouple of matches and throws them in, apparently for the same reasons, but I’m not sure that it really works.
My Mum always peirced the shell with a smallish needle, she said it stops the shell cracking. Having never had to hard boil eggs before I cannot say for certain if this helps.
Also try using whole hard boiled eggs, as a substitute for meat in a curry (add the eggs near the end of cooking just to warm them up)
for an unusual way to use up excess eggs.
Actually I believe that you add salt not to keep the whites from running if they crack but rather to increase the temperature of the water. (Adding salt to raise the boiling temperature) This knocks off a minute or so on cooking.
Update! The eggs turned out almost perfect, with just a tinge of the greenish yolk-mantle. We toasted up some Oat Bran Bread (got to stay regular, you know), added some salt, lots of pepper, and Damn! that’s some tasty lunch.