Agreed. You need to do one or the other. It's not just the digging predators. You'd be surprised at how small a gap between the bottom of the fence and the ground some of these things can squeeze through without digging. And your dust-bathing chickens can dig out a pretty good sized hole if they try.
I find it a lot easier to just lay an apron around the outside of the run instead of burying it and I think it is more effective. You don't have to bury it at all, just put something to hold it down for a little while and the grass will grow up through it and hold it down, plus the chickens will probably scratch some dirt over it. But removing the sod and putting it back on top gets the wire out of lawn mower and weed eater range and probably makes it look a little better. I suggest going out 18" to 24" with the apron, depending on how wide your roll of wire is.