16 year MMA and jiu jitsu practitioner here. The secret of success at MMA is learning how to control your opponent in ways that protects yourself. This is what the Gracie family was proving when they created the UFC. Modern fighters such as Damien Maia and Ryan Hall are taking virtually no damage in their fights.
As you learn to control your opponent the sport actually becomes very safe - far safer than snowboarding, mountain biking, etc.
Damien Maia barely deals or takes any damage, he just effortlessly takes people to the ground, chokes them out, and they tap and both walk away fine. It's magnificent to watch.
This is why I still find jiu jitsu so beautiful - it requires a basic level of skill to beat someone's ass, but it's another thing entirely to control them so well that inflicting damage becomes optional.
I once heard it described as "drunken uncle" martial arts. When your drunk uncle is being an ass at christmas dinner, you want an escalating scale of options, ranging from "suddenly he's in the living room unharmed and doesn't know why" and moving through "he's in a great deal of pain but hasn't taken any damage as you escort him out the door" to "that spiral fracture in your arm is going to take 6 months to heal". Not all forms teach that kind of finesse.
The difference is that it's suspected that boxers and NFL players actually receive more head injuries because of the padding, whereas the lack of padding on hands and heads means people's heads aren't being bobbed around too much, either because it hurts too much to hit heads as hard or because the cushions aren't acting as springs.
