I don't have a run.......the Woods house IS a covered run.
But my neighbors do.......they don't lose birds to predators. Run perimeter was built on the residual cement foundation of where an old shed used to stand.....so no diggers. Run is covered with metal roof.....which is an extension of the chicken house roof....., but run is otherwise open on 3 sides......4th side common to the chicken house. Built with sturdy (used) chain link fence tight to the bottom that prevents access to all large furry predators and raptors......and covered with chicken wire to keep small birds like sparrows and starlings out of the chicken feed. A weasel could get in, but they are not common around here.
All that combined means they do not have to shut the door to the house at night. As for the pop door........it doesn't even have one. Just a hole in the wall the birds use to come and go.
At any rate, duplicating that general concept seems to work. Covered run, made of sturdy wire, with no large openings anywhere. A physical fort knox.