Of those options I'd do rock/gravel. Hardware cloth can tear up their toes and feet, if they manage to dig down that far, weed cloth will fray over time, etc. Rocks are at least a safe option that won't degrade.
And depending on your area, wood chips may be more or less easy to get. In my...
Maybe a bit of both? We have decently soft soil, and I use deep litter which is fairly loose and they enjoy digging into that, maybe a little too enthusiastically.
Also for a while I know they were attracted to the fence line because one of the girls took up nibbling on the concrete around...
Probably the big difference is free ranged birds have the option to dig around in multiple spots, so it spread out the holes they make, vs chickens in a run where due to limited space they tend to dig up the same spots over and over.
For me the worst spots are along the edge of the run fence...
The main thing I'd be hesitant about is the tarp... in my mind it'll eventually deteriorate and the chickens will dig down and start pulling out bits of it everywhere.