The advantage of a curb bit in getting a horse to stop has little to do with pain, IMHO. Mechanically, it differs significantly from a snaffle. As you start pulling back, the lever arms (sides) rotate around the bit in the mouth. This decreases the size of the circle made by the bit and the bridle going around the poll, applying pressure to the poll. Bit evasions like stretching the head out horizontally will not relieve the pressure on the poll, nor will grabbing the bit in the teeth.
As the shanks continue to rotate back, the curb strap engages the lower jaw. At this point, the circle of bridle and bit is now as small as it will get, and I suspect it changes from a class 1 lever to a class 2 lever. The lever action then acts to leverage the horse's head to a more vertical position. Since a horse's binocular vision is somewhat limited, as is the area in their eye capable of fine resolution, it means the area of best vision is brought closer to the horse's feet - and the horse has to decide to either slow, or run where it cannot see well.
It is true some horses will continue to run regardless. The first bolt I ever rode was visiting a ranch when I was around 20. I ended up with the horse's nose at my knee, and he didn't slow a bit! He only started to turn when I started kicking his right shoulder as hard as I could, and that was pretty hard since a barbed wire fence was coming up! So yes, a horse can
bolt thru anything, as jaydee says. A curb bit, by itself, will not guarantee a horse will stop.
But mechanically, a curb bit does more to help a rider slow a horse in a straight line than a snaffle does.
I'm not sure why western curbs are viewed so warily. I know I am more than strong enough to savage a horse's mouth with a snaffle or damage their face with a bitless bridle. Since I ride a western curb with one hand instead of two, I automatically start with 50% less power...so doubling the power via leveraged shanks would only get me back to what I have with two hands and a snaffle. And when ridden as below, a Billy Allen is nothing more than a mullen snaffle with a roller in the middle:
None of this means the OP needs to use a curb. If a horse does fine in a bitless bridle, why change? If a horse does well in a snaffle, why change? However, a low end horse like Mia can do fine in a western curb, provided she is given a lesson or two in how it feels and then is ridden with slack reins most of the time.
The X-rays below show what happens with a snaffle (including the supposedly extra gentle french links) when pressure is applied:
Ideally, the horse should be trained in such a way that pressure like that is never needed. But I object to the idea that curb bits are 'harsh' and snaffle bits are 'gentle'. It seems to me a Billy Allen bit, ridden with one hand and only used when giving a cue, is one of the mildest, least intrusive bits on the market. I've stopped using the bit below because it is 5.25" across and that is just too big for Mia's mouth, but it seems to me the basic design is an outstanding one to consider if you want to minimize pressure in the mouth:
BTW - I'm inconsistent on when I take pictures of Mia in a given bit. I may or may not have finished adjusting the bridle for any given picture. However, she seems to prefer a Billy Allen about one hole tighter than a bit with links...maybe something to do with how it feels on her tongue?