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Well, you were humiliated, or I was in a similar instance, but it was different.
I had to go to the store and return what I stole (a silly key chain) as well. Seems to be a common experience. It was scary, I was ashamed, I pretty much wanted to crawl into a hole. But it had a point. I was ashamed because I had to face the guy I stole from (actually just the manager but to a kid it's the same thing), give it back, be stared down and listen to my dad apologize about me. For a kid that succkkkks.
But my parents were reinforcing a lesson they kept consistent throughout my childhood into college. Own your mistakes. If you mess up, take responsibility, face the music, deal with the consequences.
And then they taught me "move on". Seeing I was sufficiently impacted we went home, my dad cooked burgers and French fries, and nothing else was said (besides having to do superfluous yard work, raking up pine straw despite having done it last weekend, etc).
I don't know what the situation was with the girl, other than being skeptical about the parents motivations I don't have a lot of specific thoughts.
But I think there should be a "point" when parents do things, not just arbitrary pain or punishment. You hear about parents needing to exhibit "consistency", well having a purpose behind actions seem like a good thing. Humiliating a child just to humiliate them is arbitrary, I'm not sure you really reinforce anything by doing that. Making the kid return what he stole as a lesson in owning your mistakes, that seems like something worthwhile.
I think the having to return something that was stole is an effective tool that most parents use, but I am sure it was also humiliating at the same time. When I was a kid a group of friends and I were throwing dirt around from someone's dirt pile. Of course we starting throwing it at a house too. Someone broke a window. We all scattered, but our mom (my brother and mine) made us go back to the house, apologies, and offer to pay for the damages. It was humiliating, but it was the right thing to do, I learned a lesson, and I did not want to have to go through it again.
In this case though (from what it sounds like) nothing was broken or stolen. It sounds like their daughter is going through "teenager" stuff. It sounds like the parents were sick of her attitude and saw that she was not living up to her potential. Again I do not know the details. I do not know the parents or their motivation. Maybe they just wanted their 15 minutes of fame. However if this works out for the better for the girl, it is one embarrassing moment that could make long, meaningful change.
I agree being consistent as a parent is not about the punishment. It is about being consistent about the expectations and consequences, but IMO more importantly the lessons learned from those consequences. Parenting is always evolving. You change, kids change, society changes. You make mistakes as a parent that you wish you could get back. Maybe these parent look back on this situation and think it was a mistake. However I will not fault parents for being involved and trying to raise their kids to the best of their abilities. I would rather an involved parent that makes mistakes than an uninvolved parent that makes excuses.