Problems with dogs destroying grass?

Many sitters have resorted to what you mentioned, gravel, mulch or atrificial turf or pinestraw. Each solution presents itself with its own problems as none are totally dog proof. I would recommend you clean the dog prior to it being returned to the owner. Best of luck

My community and surrounding areas have lots of dirt, which can get muddy, around landscaping, and some dogs have swiffer like fur.

I've found having a plastic tub (currently using a 9 " x 13" rubbermaid) and filling with just an inch of water (With small dogs, usually I can place all four paws in at once, and have them stand in it briefly while I swish it around), and then blot dry with disposable soft paper product is the quickest, easiest, and more thorough way to keep them and the home clean. After having discussed skin sensitivities with pet parents that afflict some dogs, I may use water along with a gentle dog shampoo on the paws in a shallow sink before they go home if needed.

I feel your pain! Between dogs and bad soil and the need to conserve water (or very expensive water bills) my backyard became pretty much dirt. Eventually I want to re-landscape with a good turf, but that's just not in the budget and dirt and mud in the house was just not ideal. I got some very inexpensive do-it-yourself artificial turf to last at least through the winter. I live in CA where there is no snow but we do get rain. It's not pretty and it gets puddles but it does prevent mud! The most difficult thing is I've had a couple of dogs with tummy issues and the diarrhea does not clean up well. I do hose it off and that helps. I know it's not going to last more than a year but for me it's better than mud.