I wanted to ask yall over here where I feel a little safer...Is it not a good idea to let them graze while you watch them with the lead rope on the ground? I do, I guess because everyone else does. Sarah Lee used to jump at ropes moving so I thought I was actually helping her and it did get her used to it. Occasionally they do step on them but stop and step off of it.
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.
Many many years ago :) I watched my dad train roping horses, not for rodeos, for ranch work. He'd tie a rope to their halter and turn them out to graze (not a bunch of horses at a time - just one). He said they taught themselves to never step on a rope - never had one get hurt either. He also taught them to ground tie by tying them to a large log on the ground. He didn't tie them up short, they had 12-15' of rope. He also used to stake them out to graze. None of this is something I've ever done, but old cowboys used to do it all the time.
I used to hand graze mine in the yard all the time coz I didn't have any pasture where I used to live. It was mostly just a dirt lot. I'd sit out there on the ground and let them drag the lead in a circle around me. If they stepped on it, I'd give them the back-up signal (I use SHHHHHHH), and they learned to step off. It also taught them to back up when I wasn't on 'em.