I don't get it either...
In an arena, or any other area where a sudden spook doesn't result in a completely loose horse, I'll frequently drape the lead over my shoulder or forearm if I stop to let the horse relax, or to talk to someone, whatever. I see no problem, and if doing so makes me less of a horseman, so be it. Shoot, sometimes I might just tie the rope in a loose loop around the horse's neck, or even take it off completely and continue to do groundwork.

No comment on the bitless bridle - I didn't see "him", so I can't say whether "he" was hanging on the horse or not. Certain bitless bridles seemingly can be ridden on a contact, just like any other (English/snaffle) bit. Can't comment much on that either; I don't own a legit bitless bridle. Just halters and snaffle setups.