I think the only real solution to mud is to stop rain getting into the run if at all possible with a roof and/or shelter netting around the sides. Some of our runs have roofs, and some don't. The roofless runs look like Tom and Barbara's garden, while the covered runs look like the Leadbetter's - still pristine.
We put some slabs down around the feeders in the uncovered runs for the same purpose as you mentioned, but in pretty short order they too are swamped in mud so haven't made too much difference. I think the whole run would need paved to be effective.
I also have blocks around the perimeter of the runs primarily to prevent foxes digging under, but I am thinking of lifting them as there is some evidence that they are actually preventing the water draining out of the run. I will replace the blocks with old pallets which will help prevent fox digging and also provide a platform to walk over when collecting the eggs.