I don't have the three main ingredients: grass, leaves, and wood chips...
I just use what I have available. I don't believe you need all three ingredients. I have 3 acres of wooded land so I just have an abundance of those materials to use. Honestly, if you had even one of those materials I think it would work OK. Grass clippings would have to be spread out thinner to allow them to dry out. If not, grass clippings in a pile can clump together and go anerobic (smells bad).
Maybe try straw instead of wood chips...
A person has to use what is available. I used straw for many years, but it gets moldy and smelly if it gets wet or soiled. I find wood chips for my chickens works much better for me than straw. I would think straw in a sheltered coop would work better than straw in a chicken run where it would get rained on.
This winter I am using free paper sheds I make at home inside the coop as litter instead of using wood chips like I have the past 2 years. So far, I am really impressed on how well the paper shreds work as deep bedding in the coop. Another benefit of the paper shreds is that they will compost down much faster than wood chips when I clean the coop in the spring and dump everything out into the chicken run. Since it costs me money to haul our paper products into the town recycle center, I am better off shredding off our paper and light cardboard material at home and using it for the chickens.
I don't really care for the look of colored paper shreds out in the chicken run. When I dump paper shreds into the chicken run, I will cover them with grass clippings or leaves just to make everything look more natural. Having said that, the first good rain outside usually makes the paper shreds vanish like magic into the compost. Oh yeah, if you just dump dry paper shreds on top of the chicken run litter, they might blow into the yard. Been there, done that, too. Not a big problem, just took out the riding mower with the collection bags and "vacuumed" up all the shreds while mowing the lawn.
I've been waiting on a chip drop for 6 months but we are 5 miles outside the city so no one wants to deliver.
I did not have any luck with chipdrop where I live - about 10 miles out of town. But I discovered that we have a county landfill where the arborists dump all their wood chips. They have to pay a fee to dump the chips there, but it is free for me to take as much wood chips from the piles that I can haul out.

It only takes me about 20 minutes to manually fork in a full 4X8 foot trailer, 16 inches high, full of wood chips. If you work with
lots of wood chips, then do yourself a favor and invest in a good fork. Don't use a shovel. Here is a picture of a fork like I use.
I have both a gas chipper and an electric chipper at home, but it would take me days to chip up a trailer full of wood at home. If I need lots of wood chips, I go to the landfill. My chippers at home are good for yard clean up of fallen branches, but I don't have the energy to spend hours chipping up wood for the major coop litter for the chickens. That would be too much work for me. But, of course, all the wood I do chip up at home does get used in either the chicken coop as deep bedding, in the chicken run compost system, or used in the pathways between my raised bed gardens.
Everything gets used at home and nothing leaves the property. If it's organic, I'll find some use for it.