A correctly adjusted standing martingale does not inhibit the horse using his neck for balancing gestures except in really extraordinary circumstances involving the horse falling. Their only purpose is to keep the horse from raising his head above the effective range of bit; much like a running. One of the things I dislike about them is that a lazy, on the forehand horse quickly learns to brace on the martingale and use it for support, particularly around turns. So I don't defend their use, but I don't regard them with the horror that some folks do.
However, they are ridiculously common in the US hunter ring over fences as no more than a fashion accessory.
And yes, they are more common in the American hunt field, at least in my area, than running martingales.
As far as the OP's pony, the martingale is not restricting her head and neck over fences at all. As far as head carriage or head set, there's nothing really terribly wrong either. What could be improved is the way the pony uses herself. If she carried herself in better balance and pushed from behind rather than towing herself along on her front end, she's carry a better topline.
That flatwork would carry over to over fences as well, so she might engage evenly behind and round over her fences, rather than cantering over in a long flat stride.