We got torrential rains last weekend here also. I just started chicken keeping last fall. We had issues with a muddy run a couple times. Then I got some chipped wood and leaves / pine needles from a tree trimming company. It has been wonderful. The run never got muddy or very squishy even with the mulched tree trimmings. I was surprised. One corner of the run even acted like a collection basin before we put the mulched/chipped tree trimmings in there. It would fill up and sit with water for days. I live in San Diego and it's pretty dry usually, but some days it gets very wet. We also have times that there is a fair amount of humidity. I got the trimmings free from a tree service I saw working down the street. I just asked of they could dump it in my side yard next to my driveway. I love it and haven't had to do anything to it. It's about 12" thick in the corner that floods so easily. We also have some sandy soil some clay soil here.
I think that's what so many are missing by not having deep litter in their runs. Those soils get very compacted and have an overage on feces, so the soil is out of balance and so compacted that water cannot soak into it, but rather runs off or stands in all the craters of the moonscape or stays in the inch of soil at the top, making for a muddy, slick, stinking, fly covered mess after a big rain and then hot, humid weather.
The covering of materials acts as a protective layer for the bug and worm life that would work their way up through those compacted soils to feed on the nutrient rich water/feces and their tunneling loosens the soils, keeping them loose and more absorptive. Then the litter also lessens the impact of the foot travel on those soils, keeping them from being packed down all the time, so this further allows draining and absorbing of excess fluid. All of that cleans the feces and the stink right out of the run as the run becomes a self cleansing, nutrient rich ecosystem for insect and bird alike. No more feces sitting on the surface attracting flies as the chicken's digging and hunting through the litter covers their own feces into the compost, little by little and day by day, until you have a wonderful, working habitat for the chickens instead of a barren, lifeless, muddy, stinking cesspit where no animal should have to walk over and over for the rest of their lives.