I've always had dogs - all cocker spaniels who are very docile & family-friendly dogs, but tend to be rather greedy (which helps when training them LOL!) but they will steal food if left where they can reach it & if no-one is there to tell them 'no'.
I have never left a dog unsupervised with a child - even for a couple of seconds while I pop into the kitchen to turn something on the cooker down. I have always taken the dog/s with me.
My youngest GC is now 8, even now I wouldn't leave her alone with a dog as, despite my current pair of elderly cocker's who she's known all her life are very reliable & wouldn't hurt her, they are animals & you never know how they may react to a situation (such as a sudden loud noise waking them from sleeping) - I wouldn't take the risk.
Having said that, my dogs have been taught to leave people alone when they're eating at the table, they go to their baskets & don't beg. However, I wouldn't trust them if I had a small child eating a biscuit sitting in the living room if they weren't being watched. I am certain that one in particular could beg & may possibly decide to help herself if someone (eg: an adult) wasn't there to tell her to go to her bed & leave it alone. She's a very good dog especially with small children, she sits & bows her head, avoiding eye contact to let them pet her - but that's when in the company of a Responsible Adult (RA). But if there was a toddler eating a biscuit & no RA around.....I wouldn't trust her. The other one is so laid back he doesn't know what's going on until the other one tells him that the postie is at the door & he has a duty to bark & protect the house. 🙃
I would never, ever allow the kind of behaviour that you're describing. My dogs would be put into another room, crated or whatever.
Personally I wouldn't take the child there again. It's never acceptable for a dog to 'nip' (in other words bite) a child & if it was my child I'd be incandescent with anger. If I was the Granny I wouldn't be sorry enough - I'd be devastated & blaming myself for being so neglectful of my GC.
edited for grammar