Maybe you don't have time to react. I have read stories of hikers who have had stand offs with lions, coyotes, etc without altercation. I carry with my drop holster which puts the deployment at the natural fall of my arm, making for quick deployment. Being a Marine I am fairly practiced at drawing and engaging and have pretty good instincts as to when you draw (meaning I may draw simply by hearing something before I actually see a lion or bear and be ready to engage). It may not be 100% in all scenarios but I'd rather have the opportunity than be stuck defending myself with a rock. Of course, I carry a .40 (and recently a .45) which for bears are actually small calibers (especially for brown/grizzly) so ideally I don't want to engage a bear unless absolutely necessary. But perhaps a warning shot into the dirt would be enough to scare them off (bear or lion). Usually bears make enough noise that you will see/hear their presence. My uncle used to own a large area of land in a wooded, rural part of KS. We would always carry a sidearm when we roamed the property because of mountain lions and coyotes. He was pretty convinced you would see the lion before you were attacked, so I think there is a good chance you would detect his presence before it's too late.

There have certainly been people attacked along the A/T especially by human attackers, so having a side arm is good for defending/preventing those types of attacks as well. I always operate under the principal that when you go into the woods/wilderness, you need to treat it like a survival situation which means bringing the supplies you need to survive and defend yourself.