First of all, I think this mother, clearly needed to be take control of her child. That goes without saying. I think the dog owner in this situation could have been more forceful. She's the one that's going to have to deal with the aftermath if someone gets injured, rabies quarantine, fines, medical expenses. As a pet owner, you need to take on certain responsibilities for your own animal. My dog has always been really, really good around little kids, but still, you have no idea if she might nip or bite someone. Now my dog is an old lady who doesn't really see all that well and isn't really used to tiny people anymore, so I really have no idea what she would do if some tiny person came launching in her direction or if she might get startled and snap at some hand that comes at her just to pet her, so when I have her out in public, I'm taking extra precautions. Overall, I don't have a lot of worries, but if some little kid is mauling all over the dog and the dog is unpleased with this, I think this person should have and could have spoken up, to the child, and also the parent.
As a once-upon-a-time veterinary technician, I can relate to the OP's dilemma. You do the best you can to assure the safety of the pets and people, but we're not running a daycare here. She was running in six different directions and didn't even notice the child to begin with. I have been that person trying to get the food and the prescription refill and answer the phone and pull the chart and certainly once you see the dog and the person in distress, you're jumping in to correct the situation. You expect pet owners to know their pet's temperament. I think the dog owner should have been more vocal with the child, with the parent, and also to get aid from the tech if she needed it.