Mountain lion attacks against humans increased dramatically after Calif. discontinued legal hunting of them. It's better for the cougars if they don't lose their fear of us, and it's also safer for us if they are afraid of us. From everything I've read, the wildlife management people say there is a very healthy population of them in AZ. I agree with continued limited hunting. Without dogs the success rate is about zero for hunters, so only the people with the trained dogs can usually find them and tree them. And there are not huge numbers of these people, there aren't a lot of lions being hunted in the state, compared to any other game. And if cougar hunting suddenly became very popular, the G & F Dept. would probably respond by limiting the tags. This is not likely to happen, as most of us do not want to be saddled with caring for 6 or 8 very large, noisy hounds in the backyard!
As for why anyone would want to hunt them, I'm sure some people want the skin, or for taxidermy, and they are also pretty tasty, believe it or not. I tasted the meat one time at Tucson Rod & Gun Club's annual game BBQ. I personally would not want to hunt them, however. I like the fact that there are large, scary predators running around out there, ready to pounce on unsuspecting city slickers.