My flower beds are improved beds and have 8" of good soil on top. Even there, their "holes" are only about an inch deep.
Digging down 6" just seems crazy but i guess i have lazy birds.
Probably the big difference is free ranged birds have the option to dig around in multiple spots, so it spread out the holes they make, vs chickens in a run where due to limited space they tend to dig up the same spots over and over.
For me the worst spots are along the edge of the run fence, they really like to excavate those areas. In the spots where they've dug down the most I placed large branches (like roost size) across in the indentations so they can't dig right up against the fence.