Emergency stop? This is the ability to stop a horse dead in its tracks reguardless of what is goign on. I am not talking about every day in and out stopping. For that training is the only way to go.
What I mean is when your horse has become so hysterical, for whatever reason, he is running full out and about to get you both killed. Types of things that could cause this for most horses:
Explosions
A mountain lion/bear (seen two of those in the last decade)
A tree falling
A dog tearing chunks off your horse's legs
An son of a "gun" who tosses a glass of liquid in your horse's face as they drive by
However, for a reactive horse, this list gets a lot longer...
Cows that he swears were bears
Buffalo
Chainsaws
That occasional odd looking leaf
Or, in the case of my horse, landscaping rocks
You get the idea. The point is that every once in a while - hopefully not very often (once a decade is more than enough for me) - things go terribly terribly wrong. No matter how great the training of a horse is, there are times when they are too scared to be able to think and all they can do is RUN.
If you are riding in a mild bit that normally takes medium pressure to get the horse to respond, you are dead if that animal gallops out into traffic or off a cliff, etc.
Now, I ride with a harsher bit then most. I do not engage the bit AT ALL 99.9% of the time in normal riding. My horse works off of leg pressure, seat, and rein pressure on his neck. Should the worse happen, and it has a few times in the last 12 years, he can be very reactive, I can engage that bit and stop him dead in his tracks FAST. Wreck avoided.
I see a lot of people who love to say that they can ride in a "mild bit" wind up hurt because although it works out fine in normal situations, in an emergency, they get hurt because they dont' have that extra bit of control.