Leghorns can be exceptionally good flyers for chickens. I read a post years ago from someone who thought her chickens had all escaped or been stolen, only to look up when she thought she heard something and discovered they were all roosting in the VERY tall tree next to their run

They are very good layers, and very good foragers, and always alert to any potential threat - they are considered flighty because they basically think everything that moves is a threat. They stay alive that way.
So, the way to accomplish things is to always close the gate into the run immediately after you get through it before you try to do anything else. If there is ANY chance it can latch in a way that traps you inside, fix that first. When you go through that gate they will almost certainly go to the other end, under or into the coop, especially at first.
If you can make your fencing around the run 6 feet tall with a cover, so you can walk in without crouching, and keep them inside, you will be happier, and it looks like that's actually what you've got, if I'm understanding your description correctly.