The purpose of the noseband is to stabilize the bridle, which might need it because we have direct contact with the bit.
In Western World where the finished horse works from a signal bit, much less moves, so in theory there's no need to stabilize the bridle on the head.
And the well-trained Western horse wouldn't dare rub his bridle off.
I think the one-ear contraption was there to let you see a big cheek and a small throat latch bred into that pretty horse. The two-ear rendition is probably about symmetry and one more place to put some sliver bling.
In western, some might use a western noseband during training with a tie down (essentially the western version of the standing martingale).
Otherwise for english, the noseband can be both functional, and non functional depending on the horse and it's use. But if one needs to attach a standing martingale to a bridle, a noseband of some type is required.
A one earred bridle can be VERY dangerous. Multiple times I've seen them slide off the ears and the side of the bit go into the horses mouth when somebody pulled hard on just one side of it (as happens with green horses and/or green riders). Plus some of them can't be comfortable for the horse, they often seem too small or shaped wrong and leave very little room for the ear without rubbing. And I think the doube ear ones are just plain fugly.
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They probably don't allow their horses to rub on trees or posts, but then again, neither do I.
And neither do I, but poop happens and sometimes dobbin rubs his face on his legs before I can catch him. I don't like the fact that the only thing preventing the bridle from sliding off his head is a little loop around the ear.
He has a one eared headstall for western with his curb bit. The curb sits in his mouth and is used such that twisting and moving around aren't really an issue since its a signal bit, not a direct contact bit. The headstall doesn't work well with a snaffle bit unless you put a curb strap on the snaffle also. That's generally how snaffle bits are kept from sliding through the mouth.
I've never worried about him getting the bridle off, even as a confirmed head rubber. Even if he did rub it off, he's well trained enough that it wouldn't really matter.
Even on my English bridles (one for dressage and one plain traditional one for foxhunting) the nosebands are adjusted so loose that I don't undo them to put on or take off the bridles. He gaped his mouth and did all kind of weird things when he was a baby, but grew out of all of it and now just softly mouths the bit like he's supposed to.
I've been told they started using nosebands for hunting because they prevented certain injures to the horse in the event of a fall. If the horse didn't have a noseband on and had its mouth gaped open, the lower jaw could impact the ground and break. If the noseband holds the mouth just shut enough to prevent that wide gaping, the horse is less likely to break its jaw. It won't prevent all injuries, but it could prevent that one.
I use one ear headstalls and prefer them. Easy to get one and mine stay put because they are adjusted properly. A ring snaffle has a snaffle strap (too early for me, can't think of official name) which goes on the rings, under the chin, so you can't pull the ring through their mouths. I like the earpiece to be the slider kind so it can be put in the correct place, not the ones which are cut into the leather. Good quality leather is the order of the day. I'm not really a fan of throatlatches. Something to get caught on brush or something.
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From what I had always heard, the one ear headstall is from the old school Californios and Vaqueros. FWIW I ride one horse who loves to rub his browband bridle off, but has never even tried with a one ear. He's much happier without a browband and throatlatch.
The western bridles I've seen with the ear loops have no throatlatch. The horse can take them right off with a good rub against a post or tree.
The more I look at western tack the less I like it. All of it.
Don't knock it til you've tried it.
It is generally a pretty bad idea to let 'any' horse in 'any' bridle rub its head on post, tree, adjacent horse or rider, what have you. I've seen some pretty good wrecks that way (in English bridles).
I've been using one-eared, no throatlatch or throatlatch bridles since the 60s. Works just fine for me. I do have a friend who can't use them because her horse does rub them off- but happily I have not had that problem over the decades. I do use a browband on the gelding lately- loaned him to someone who was not careful about putting the ear through the ear loop, and the particular horse has aural plaques so her one bridling session caused him great defensiveness that is still, months later, not entirely gone.
I often use the one-eared bridles when riding English as well. less leather to clean.
My mare requires a headstall with a throatlatch. I used to ride her in a headstall with just one ear, but one day on the trail she taught herself this nifty trick where she sucks the bit up in to her mouth to loosen the crown piece then gives a good bob with her head and flips the headstall right off.
When I use a snaffle bit with my western headstalls, I always put a bit hobble on them to keep the bit in place.
Many western trainers use nosebands with their western headstalls when starting horses. And as was mentioned earlier, horses used in speed events often where them along with a tie down to help the horse balance itself during high speed maneuvers.