I agree with Dream and Toots. Yes, when you pay for a meal, you are paying someone else to prepare your food and do the dishes. However, I think that in an establishment that is not specifically marketed towards small children, it's not unreasonable for proprietors and other guests to expect that parents monitor their children's behavior. That means removing them if they become unruly and making sure that they aren't throwing their food on the floor. Of course accidents happen, but I do think that the mom in question probably could have prevented that many bits of scone from winding up down there. As Toots said, I'm not sure what stopped her from scooting his chair in closer to the table.
Finally, the customer definitely outed herself. Had she not admitted that she was the customer in question on Yelp, nobody would have known. The photograph didn't mention her by name, nor did it show her or her kids. This really isn't much different from someone posting a picture of a car double-parked (cutting out the license plate).