I don’t think that the point is to have them out and able to be seen by the judge, but I think that the items in question should be available to be gotten to quickly.
My horse wears a rope halter under her bridle and I have the lead rope attached to one of the rings on my saddle right behind my right leg so I could grab it quickly enough if I were to hop off and want it.
Lead ropes can be very useful on the trail, on not just for leading… this weekend I went to two rides at Mammoth Cave Nat’l Park here in KY. On the Saturday ride, a horse a judge had ridden out was standing tied to a tree out of the way and a sinkhole opened up under his hind end! His hind end was in the hole, one leg kind up on the side and the other dangling. The horse could not get any purchase to get out on his own (luckily he was calm). We got a person on his head, a person on his tail, and myself and another guy got a lead rope around the dangling hind leg- with all four of us pulling (head, tail, rope around leg), the horse was able to get enough leverage to get out of the hole… it could have ended badly, but luckily it did not. But without that lead rope, I’m not sure what we would have done!