I think for just that reason.... Most English riding is all about being soft and supple at the mouth, poll and neck, always giving to pressure and seeking a round-necked frame. There is no English situation I can think of (except maybe polo) where you ever want the horse to lean on it, as there is in those western events you named.
In English riding situations with a horse who will throw his head or carry it too high, English disciplines prefer the passive restraint like draw reins or running martingales as a training device, which are only ever in play to correct an out-of-position head, and not in play when the horse is in proper position (running martingale), or as in draw reins, to use leverage to encourage the softening at the mouth and poll. Some jumpers use tie-downs, but I wouldn't use them there, because I want the horse to have full use of his neck if he were to stumble on landing.
But if they are leaning on the tiedown, vs. the bit for an aid to get their butts under them that DOES how keep them softer in the mouth. When a horse is draggin a cow or making as tight a turn as you ask them to going around the barrel at the speeds they do, you have to give them an aid for the right postion. It will be a tiedown or their mouth.
I can't think of an English Event (except polo as you say, and I've seen alot of them with tiedown,) that requires a horse to drag a cow or make hairpin turns at full runs:). Barrel Races and team ropers want their horses soft in the mouth and to give to pressure. The tie down is an aid to help do that.
Becky