I would never leave a horse in a stall in a non-breakaway halter. I use all nylon for everyday use, but those are only on him when he's being led or groomed/tacked, and tied with a quick release knot. I would not leave a halter on in a stall, pasture, or trailer that is not breakaway.
And yes, they do work. I was trailering my horse in a leather halter, and upon arriving at our destination my non-horse person dad thought it was a genius idea to open the trailer door while my horse was still tied. Horse started backing out and pulled back when realizing his head was still stuck, and all it took was a couple of good pulls for the halter to snap in one place (where the crown piece connected I believe) and he was just fine. If you have an all leather halter break there isn't really any way to continue using it, but the ones with the breakaway tab under the buckle or the breakaway crown piece can be fixed. You can just buy a replacement tab or crownpiece and it's good as new.