Not true that they are intended as a training aid. Tho people in the UK may get them for that purpose.
Rope head collars are standard in some areas of USA - largely in those States where leather was in short supply or cost too much.
The YO where my share was kept switched to rope halters and everyone followed suit. I used to lead my share out to grass in whatever head collar she had on and tho in theory the rope was said to give more control, I didnt notice much difference in my control. Or communication. Body language has a lot to do with it.
The horses were tied up in them too. To bailer twine of course.
Whereas yesterday on another yard I saw a heavy horse break and wreck a good leather head collar.