It's a tough situation: a child behaving highly inappropriately in public. I would not want my child to be allowed to do so with impunity, but I would also not want my child to be disciplined in ways I consider inappropriate--and, of course, ideas on appropriate discipline vary quite widely!
I would suggest that verbal admonishments should probably be used more frequently--"You need to stop shrieking." "Hey! Don't run inside the building! That's an outside game." People (including me) are reluctant to speak up, quite naturally, but it seems to me quite legitimate nonetheless. And I wouldn't threaten in any way; simply stating the rules and expectations that apply would normally be the limit.
(I wouldn't attempt to reason with a younger child, either, by the way--not because it's inappropriate ethically, but because it's completely ineffective. It might work for a teenager, or a preteen, if you're lucky. It surprises me how often people try elaborate ethical arguments with a 6-year old, but I've never known it to have the slightest evident impact on the child--other than perhaps to bore them.)
On the other hand, I wouldn't ever touch someone else's child, except possibly in the most extreme circumstances, such as stopping an actual continued assault by the child. (A circumstance I've never seen in public, thankfully.)