The most important thing about safety reins (their proper name, BTW), is that they not only protect your child in situations where your attention is momentarily distracted, but in situations where the adult may become disabled. The poster who mentioned her mother tripping as they left EPCOT is one example, but I have another, even better, one. (I've posted this story on the DIS before, but it bears repeating for this discussion).
Safety reins have been used in Europe for centuries, and they are still pretty standard in cities whenever people go out on foot with young children in areas that have heavy traffic. This particular incident happened before I was born, in 1944 in fact, but it is still quite relevant. My mother was going out to do her grocery shopping, with my then 4 yo sister walking with her, wearing safety reins, and my brother, who was an infant, in a stroller (more specifically, a big, traditional pram.) The street they were walking down was quite steep, and they were going downhill. The traffic was very heavy. It had been raining, and as they came to a crossroad, my mother tripped as she stepped off the curb, and her feet went right out from under her. Her left leg broke, and she struck her head on the curb and knocked herself out. Naturally, she lost her grip on the stroller, and it careened into the traffic, gaining speed as it rolled downhill. My sister, terrified, tried to break loose and run after the stroller, but she could not because the reins she was wearing were looped around my mother's wrist. Cars started crashing into things as they swerved to avoid the stroller. Bystanders summoned an ambulance to take my mother to a hospital, but in the confusion no one realized that the baby in the stroller belonged to the lady who was hurt, and by the time the stroller was stopped and my brother rescued, it was several hundred yards away. As he was (miraculously) unhurt and no one was claiming him, my brother was taken to a police station. My mother, by this time awake, was frantic, of course, and my father was found at work and sent to try to find the baby. It took him several hours to discover what had happened to my brother. The bright side in all this (aside from the fact that no one was hit by a car), is that because of the reins, my sister stayed right by my mother's side, safe.
I've tripped and fallen at theme parks, and I once fell down the entry stairs of a busy London Underground station. It can happen; you *can* get hurt when you are responsible for your small children.