Our rope halters have the lead permanently attached-part of the idea of a rope halter is no hardware so the leads are always tied on.
I never ever ride with the halter on under the bridle-too much junk on their head, just more stuff to catch on other stuff, more stuff to get itchy/hot/tangled/out of place and it just looks dude-ish to me. The lead rope left on the halter and tied back to the horn really bothers me, I’ve seen several wrecks and near disasters from horses tripping and stepping in the lead or catching it on passing brush or in one memorable steep mountain trail occasion the horse in front kicked the horse behind and got a full leg between the lead rope and the horse. lead rope was tied on hard with two half hitches-big mess.