This is just something I've noticed with my horse (and probably a result of his badbadbad former training before he came to me...) but just putting my reins through a running martingale without engaging it even once during a ride makes him OODLES softer in his face! He also engages his back more at the canter, oddly enough. He's a weirdo, though, so I wouldn't call him the norm.
I'm not sure how a running martingale would "save a horse's mouth" on principle, though... how odd! I don't particularly love any contraptions (outside my bit, bridle, saddle and girth!) to begin with.