There's a risk of them wrenching their neck if they step on the line at the same time they startle or jerk their head up, but really, it's a small one. Some horses panic about the line getting stepped on or tangled around feet more than others. If you're right there to correct things, I'd do what you think is safe.
I'll have to think about this. Too bad about that horse though. You ever see that John Wayne movie, I can't think of the name of it but his daughter coming home from school and he and his wife spliting up. Anyway there was a scene kind of like what yall were describing.