I have a mulberry tree in the run. During mulberry season the first place they go in the morning when I let them out is to see what fell under the tree overnight. It’s not a problem.
Don’t be surprised if you see some color in the chicken poop. I feed mine beet skins when I can beets. The first time I did that I thought they were pooping pure blood until I remembered what they ate. When I feed them red cabbage leaves you can see an interesting blue sheen in their poop. There is nothing wrong with this, it’s normal, but it can be disconcerting if you are not ready for it.
With that many trees you might consider making mulberry jelly. That’s one of my favorites. Strain it to just get the juice and avoid the seeds. Some people don’t like mulberry trees because birds eat them then poop on sidewalks, driveways, or cars and stain them, but my tree is far enough away that is not a problem. I like my mulberry tree.