A break to the neck won't always kill an animal. There's a horse in the UK with a neck so broken it protrudes at a severe angle and she can't lift her head, but she's alive. I've also seen a picture of a giraffe with a similar break. His neck was twisted out to the side, obviously broken and distorted, but he was alive.
I don't need to go to a rodeo to see bad horsemanship, I just wait for the local fair. I swear... I used to enjoy it when I was a kid. Then I showed there once and it really opened my eyes. I could tell you many, many stories.
I've been to a couple big name rodeos and I don't remember anything bad happening. Livestock is expensive, especially the good ones. You don't want it ruined in one event so they try to keep them happy and healthy. I've done pick ups for some lower grade rodeos and never saw anything bad. I'm guessing the ones without much regulations who host unskilled parties get the most injuries.