There's a difference between a horse who takes the bit and runs through it and one who *can* take the bit when you misjudge your stride or something happens like a trip or something before a fence where the horse needs to sort itself out. In those situations, you want a horse to be able to help you out, but of course you don't want something that is barrelling through the hand all the time when jumping. I jump my horse in a Cheltenham gag when out (as that's when he gets excited and strong), just so I have that bit extra ability to hold him when he gets strong, but its not so strong that he can't take hold of it if he absolutely needed too. From what I've seen the double bridle is too much for a horse to take hold of in that situation, which is fine if you want to be the absolute boss and have control the entire time when jumping, but especially cross country I like the partnership and the trust of knowing you will help each other out when you need to. Especially in cross country, too much control can be a bad thing...
It stems, I think, from an incompetent rider in his past. He has somehow connected 'no contact' with getting yanked in the mouth over fences.