I would imagine that they use the running martingales in Endurance for much the same reason as they are traditionally a part of the standard fox hunting tack in the UK - they are a great safety net if you need it (and you never know when you might) but as long as they're fitted correctly don't otherwise interfere with the horses natural head carriage
I once heard someone say you can tell where a horse is looking by the head elevation. And, yes, Arabs do have a naturally higher head carriage than many other breeds.
When the horses out in the pasture bring their heads up they are focusing on something in the distance. Same on the trail. If that head come up they are watching. The head is much lower if we are walking along and the nose goes down if they are watching the ground for a puddle, tree roots, whatever.
I would expect an endurance horse going down the trail at a trot to be looking in the distance a bit. I did LD a number of years ago and to me it seems most of the horses traveled as naturally as possible without being hollowed out.
I use a running martingale on my horse when I ride with others or in a new place. Reason being is that my horse will sometimes try to evade the bit by sucking back and twisting his head. It's near impossible to do much besides hang on when he does that without a martingale. With the martingale it's much easier to send him forward and bring him back to me. Posted via Mobile Device
"The horse also has a "visual streak", or an area within the retina, linear in shape, with a high concentration of ganglion cells (up to 6100 cells/mm2 in the visual streak compared to the 150 and 200 cells/mm2 in the peripheral area). Horses have better acuity when the objects they are looking at fall in this region. They therefore will tilt or raise their heads, to help place the objects within the area of the visual streak."
Also, how a horse uses its back is much more complex than the simplified rules we tend to apply. These short articles give a good taste of the difference between what we normally discuss and what actually happens: