Hmm, i sure enjoyed Chullicken's post, but have a very different take on part of it.
Here is why area of the country matters.
We are on a base of sand (sandhills of carolinas). The real sand in our cleared areas is nasty. In the pastures (7ish acres fenced) there were previously run crops - different ones that do well in sand base, poor soils. Then a few full size horses, then a few minis. When it rains here, it smells bad - even in open/cleared areas (no animals). When it doesn't rain, any watering of plants runs straight off. Even w/ no humidity (actually a bit rare), both the dogs or cats roll in the sand - they smell BAD. A brushing will remove the scent, but...
I have watched other chicken keepers purchase sand to put into their runs. And in short order, I've seen those same friends & acquaintances remove the purchased sand from their runs. Removing stinky sand is much harder than hauling in the bags, LOL.
Just about everyone i know has gone to either a deep mulch of wood chips or actual DLM.
As to products to use - i find you don't need to sweat it or overthink it. The wood chips WILL break down & disappear. Thats what you want as it will add drainage to a clay based, non draining soil (in ours, it adds good material that helps to retain water & nutrients). So you add more material, as needed for digesting the waste & pretty. Since it appears your coop run is in the backyard (seen more readily), you may not be able to use all the different materials that i have incorporated, but it is food for thought.
Our
DLM
In the future, you can always roof the run. You may find, as you get older, you won't want to shovel so much. Keep in mind, too, that some chickens just do not like going out in the snow, but you can lay down straw in their run that may entice them out of the coop.
One of the things you learn during "chickening", is that if something you do doesn't work for either you or your situation, you can try other methods.