Seconds count when you are confronted with a animal meaning to take you down.
I've seen quite a few wild animals out riding and never felt threatened by ANY of them. Could it happen, sure. But I'm much more likely to die of cancer or a car wreck. I think the risk of wild animals is wayyyyyyyyy over-rated.
I've seen bears 3-4 times........they always ran. Now those are black bear, so grizzlies are probably a whole 'nother kettle of fish! But we don't have those in Arizona.
I've seen badger, rattlesnakes, coyotes, deer, elk.
I love to bugle for elk on horseback during the rut. Last year I was in a hunting store and when I asked the employee where the elk calls were and told him what I was doing he told me I was going to get "horned!" Well, anything is possible I suppose but I've been enjoying the rut by bugling for elk on horseback for 20 years now.
Even the bulls, once they see you are not an elk, are out of there! I've had some cute little elk calves run up within about 20 feet of me though......at that point you are more worried about your horse spooking.
Rattlesnakes have such a bad wrap but I've never had one bother me and I've lived in Arizona my whole life. If you keep your hands out of dark places and keep your dogs away from them, they are generally no problem. They could bite if you are in the wrong place at the wrong time but generally they are peaceful creatures. I always thought it polite that they rattled......like "hey, I'm hear, don't step on me." Most snakes don't do that.
Coyotes are pretty cool. I've had them "shadow me" and follow me on horseback and in the city I saw them following bicyclists. But never saw them as an actual threat. Now for sure they would go after calves or other small livestock. So for a cowboy they could be a problem......not for the cowboy but for his livestock.
No feral hogs or gators here, thank goodness.
I had a javelina circle around me and my horse once, grunting and breathing heavy. That was pretty creepy! Oddly, the horse didn't care. Which was great, because I was a bit worried about it! My Dad, who has hunted them, said their eyesight is poor and it probably had babies to protect. I rarely ever see javelina, and that was the most memorable encounter.
Anyway, the risk of wild animal attacks, at least in Arizona, are pretty remote. Driving a car is much more dangerous and people don't seem to worry much about that.
PS. I've never seen a live mountain lion, I would love to see one! Saw one that got hit by a car once and it was gorgeous.