But if lack of punishment of some sort is the reason why he is acting like that, then would not acknowledging him or taking him home be effective?
Lack of punishment earlier in the child's development is what I meant. Once a child gets to the point that this child is at,things have to be done differently. If this mother would have consistently punished past behavior appropriately, it would be unlikely the child would be acting this way. Since the child is acting this way, then different measures have to be taken. This child is making a scene in a public place to get what he wants. He has learned through his development that making a scene will embarrass his mom in public and she likely has given into him in the past enough times to reinforce this behavior. At this point, by not acknowledging him, she would be taking away the one thing he is using to try and get what he wants: causing a scene. Spanking him in public is still causing a scene and plays precisely into what he is trying t do. If she spanked him,he would likely start screaming and continuing to do exactly what he is already doing. If she doesn't give in to him on a consistent basis when he throws his tantrums, he will learn that they are ineffective. Also, by removing him from the situation by taking him home, she is showing him again that his tantrums will not result in him getting what he wants.
I am not defending this annoying brat's behavior, but the child was not born this way. The child developed these habits through bad parenting. Simply hitting him in public is not going to force this child to behave because he is still accomplishing his goal of embarrassing his mom in public in hopes that she will give him what he wants just to shut him up.
So when I said lack of punishment, I meant in the child's upbringing, not in this setting. This child is way past the point where a punishment administered to him in a store is going to make a whole lot of difference.