As `shintakezou pointed out, this is very much a cultural matter here. France (where my parents and I are from) tends to subscribe more to the "it takes a village to raise a child" mentality. Correcting the behavior of someone else's child at the park, supermarket, in the bus, etc. if they are out of the parent's sight is perfectly acceptable in French culture, and as a kid it happened to me many times, without traumatizing me (even when it was a scary old man yelling at me).
It is not perceived as assault, and it is certainly not "terrifying them verbally" (I find that phrasing a little bit excessive though, and am surprised that you extracted that from my "a few stern words").
In retrospect, the one thing my father did wrong was not considering that he was living in a different culture, and that something that seemed normal to him might be scandalous in the US. But that only makes my wish to leave the US when I have children even more valid in my eyes: I'd hate to find myself in those sort of cultural misunderstandings that can very easily escalate juridically.
I do find the US's approach to parenting very poor: France's approach to parenting is that kids are human beings in construction, and that they do not inherently know what is right or wrong, good or bad - and it is up to adults (parents or not) to let them know (that doesn't mean that their feelings or interests are invalid because they are "incomplete": it just means that they are lacking the knowledge of social mores, conventions, mental tools, etc. required to be productive citizens). The US's approach to parenting falls more in line with the belief that children are just "little adults" and that one should respect their wishes and desires no matter what, which I find preposterous.
scolding != assaulting, no?
But of course it might have been better if the kids’ parents took up the matter between them.
Once common sense would have been enough: do you scolded my child? Why? What's happened? ... Ha, ok, my son did something wrong... and so on...
Usually it's not that hard.
Instead, a society rambling towards paranoia and driven by fear (especially this), makes a citizen feel the urge to "look at" a greater power, the Great Protector, to "protect" him/her even from an unexistant menace - or to seek revenge.
In the story we are commenting, the first thing I have noticed and blamed was the following: the mother surrenders to the request of the son who wants to go with her to the store. There, she makes the first mistake: the son had to stay at home with the grandma.
When we lived in the US for 2 years (pre 9/11), my parents definitely experienced a few police-state-related cultural shocks.
One time, a kid started hitting me in the school bus. It turned out that the kid lived in the same apartment complex as we did. That evening, my dad saw him playing outside, went up to him, and gave him a few stern words. A few hours later, his mother had called the police, and my dad ended up spending a few hours at the local police station.
We still lived in the US for another year or so after that, but nowadays my parents heavily dislike this country, and tend to avoid visiting me as much they can. FTR I don't see myself staying in the US when I start having kids.