I am a criminal justice major
The role of law enforcement in a hostage situation is to slow down a situation and contain the chaos. Running is not the best solution since you put the safety of the hostages at risk. Law enforcement can only shoot people if the totality of the circumstances is met. A gunman might have a weapon pointed at someone with the look that they aren't backing down. You might order them to drop the weapon, but they don't and continue to brandish it, they might shoot or charge you.
Shooting to wound is a bit harder than what you guys think. It's really easy to hit the center of mass. Unfortunately that is where the heart, lungs and stomach are, pretty much the best place to Arms and Legs are pretty hard to hit, even if you do hit them you might not even break the bone. The weapon caliber can also play a role bone damage. Another thing to consider is that not everyone's body is built the same or has the same chemical composition. A gunshot that sends a man falling down screaming can also be a gunshot that leaves a man standing still charging at you.
Arms and Legs are pretty hard to hit, pretty time consuming to aim at. Look at these sights for example:
In city near me there was incident where an officer shot at a gunman with an AR15, the gunman took a hit to the heart and proceeded to move forward. From the officers point of view it looked like he was charging forward, so the officer shot again and hit the dude in the head. Split in it open like a melon. What really happened was the dude got hit in the heart and was stumbling forward to fall over. A lot can happen in a few seconds which require split second decisions.
Pretty much ending the whole thing as bloodlessly as possible is the goal.
