It reminds me of many occasions we've had with our daughter. The mother stayed very calm and I think she handled it well, even managed to smile and not raise her voice. He looks like a very heavy boy. If he's anything like my girl, he's much bigger and stronger than his peers and strapping him in would be near impossible, in the middle of a tantrum. So, I fully understand why she appeared to give up trying, until he had calmed down. As for the dummy (pacifier), my daughter never had one, but she sucked her thumb a lot until she was about 5 and still does if she's very upset, but not very often (she's 6). Yes, it was a comfort to the boy and keeping it from him might seem cruel, but as far as I can see, it did the trick. He went into his seat after that. I bet he fell asleep on the way home. He seemed just like how my daughter would be if she was very tired, possibly over-stimulated.
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"We act as though comfort and luxury were the chief requirements of life, when all we need to make us really happy is something to be enthusiatic about." Charles Kingsley