Yeah, for your run, you do not need to have any sort of bedding.
I currently have two runs.
One has a gravel floor that my DH poured in thinking that would be "best."
I have bags and bags of raked up leaves that I use to dump onto the gravel (I hate gravel).
My chickens free range once I get up in the morning to open up the gate to their run.
My other run is inside my shed, and I use raked leaves and mowed grass as bedding in the run.
Before we moved I had a chicken coop where the run was outside on the grass. That was so nice. No litter necessary.
Then the chickens scratched up all grass and it became a mud pit. So I started filling it in with the mowed grass clippings and raked leaves.
The yard for your chicks looks nice!