This falls under the "greater danger" theory in that it's more dangerous not to act than it is to act.
scenario one:
Bad guy goes into a mall and starts shooting. Already killed 3 people when an armed customer shoots back. Armed customer hits another shopper before hitting bad guy. Score: 4 innocents dead,1 bad guy dead.
Scenario two:
Bad guy goes into a mall and starts shooting. Nobody shoots back until police arrive 7 minutes later and kill him. Score: 1 bad guy dead, 40+ innocent shoppers dead.
Yes in #1 the armed customer hit an innocent. But if no one attempted to stop the BG a lot more would have ended up dead. It's more dangerous not to act than to act!
Very good presentation of two choices, and two outcomes.
Fortunately, it has been proven, that Americans are capable of making those kinds of tradeoffs, and accepting the results. Unfortunately, there is a tragic learning curve, unless you have already accepted that you may be in that type of a situation.
We are not talking about chasing a robber thru a mall, in this example, but rather dealing with someone attempting a mass murder.
-------
On September 11, 2001 the worst incidents of mass murder in the United States took place. By the time two planes had flown into the Twin Towers, and flight 77 hit the Pentagon, the people on flight 93, realized that they were in a new type of event, not simply a hijacking going to Cuba/whatever or a Hostages for money/notoriety/exchange situation. It was a mass murder of all the people in the plane plus whoever they could kill on the ground when the plane crashed. The passengers and crew, changed their tactics, realizing actual human lives would have to be sacrificed short-term to reduce the amount of casualties long-term. They did have a hope of preventing the crash, but many feel that the action would have been taken, even if they were certain all the people on the plane were going to die.
It went down in a field in Western Pennsylvania, wih no casualties on the ground. I'm certain that the decision to rush the cabin and try to regain control of the plane, or force it down NOW, was not unanimous, but those who have studied what we know of that situation, are universal in agreeing it was the right thing to do.
Opposing a heavily armed foe whose goal is to kill as many as possible, is a warlike situation. It demands aggressive immediate action, and is likely to have direct and collateral losses of innocents, especially when those innocent third parties are overmatched in their ability/resources to defend themselves. The use of human shields makes this situation even more difficult. A win/win outcome is very unlikely.
You may need to engage to the death, to stop the losses. People who do this in wartime are called heroes. Of the few living heroes, most of them give credit, to those comrades who died in the process, and made it possible for the hero to survive.
I wish i could express this better, hopefully you understand.