Can I just say - All of you are talking about the head being 'restricted' - If a tie down or martingale is being used correctly it shouldn't restrict the head at ALL unless it is almost vertical and about to hit the rider in the face.
A friend has/had a pony who is an awesome trail horse - Super brave, always our trailblazer, goes all day. He also has a super short neck and anxiety issues with a bit - He does occasionally do the head flip, so she has in the past trail ridden in a tie down. It was NEVER short enough to effect his balancing one iota - it only became taught when he tried to head flip.
My Arab is ridden in a running martingale most of the time on trails because when he gets excited, he head flips. The martingale on him is ridiculously long because he doesn't even try when it's on. We ride some pretty rough country and again, it has never affected his balance or head carriage.