I like the question, it shows you are thinking.
Some of us don't put anything down. If you have enough room the grass can remain and the poop is really spread out. If it drains well runs can stay pretty dry even when it rains.
But many runs become a muddy mess when it rains. That can get things dirty and can be unhealthy. Drainage is important, but especially in a smaller run some type of ground cover can help keep it less muddy and drier.
Another issue can be that the smaller the run the more the poop builds up in it. Poop can get so thick that it stays wet and will stink. If you have a lot of area the poop will be spread out so it's not so much of a problem but many of us do not have that much area. Adding bedding helps the poop dry out and be less of a problem.
Some people add sand or gravel so water can drain through it and the run can stay drier. That only works if the water has some place to go. If you dig a hole in clay and fill it with sand you still have a wet pool of water, it just has sand in it.
My run is on a bit of a high spot so it drains pretty well. I leave it as dirt, no bedding. But when it sets in wet for a while it can get muddy in places. So I spread pea gravel in areas I want to walk or stand to keep me out of the mud. The chickens have some higher spots and grass where they can avoid the mud too if they want. My run is over 2,000 square feet.
