I wanted flagstone at the base of my stairs, but decided not to use it because of what Jax mentioned above. My dog jumps off the stairs, and even though I might not like the look of the dead grass and the mud, that's what I'd rather he land on - instead of hurting himself on slippery (and beautiful, sigh) flagstone.
My method of keeping the lawn in 'reasonable' shape is to either reseed with white clover, or in the extreme dead zones, just resod the area. Even the sod that doesn't take will keep the mud covered, and the grass will come back next year.
Or do it up in flagstone, have a gorgeous fire pit area, and then put some chairs back there so they can't race behind it? But then they'll just tear up another spot, lol. Dogs are a nice lawn's natural enemy, face it