The primary value of catching the "perpetrators" is deterrence for others.
The primary value is stopping further activity by the perpetrators. The secondary value is deterring others. By treating this as a war, we have not only failed to take out the perpetrators (remember that guy, what's his name.. oh yes, I remember now: Osama bin Laden), we have recruited many thousands of fighters for al-Qaeda and related groups. When you engage in police activity, you target the perpetrator. When you invade two countries and engage in military operations in several others, you turn people who would otherwise be bystanders into combatants.
Putting terrorists in jail will not deter those in the future - they are already willing to die for their cause, no threat of punishment will prevent them from going ahead.
What cause? Initially they had a small cause. Now we have made it a much larger one. As for threat of punishment as a deterrent, you are assuming that all terrorists are suicide bombers, which is definitely not the case. Suicide bombers make up a third of the people who engage in terrorist acts across the globe.
So the idea that you are going to identify the "criminals" and put them in jail/execute them presumes that you will just take the hit, no matter the cost, and deal with the aftermath. That's why the "policing" concept has utterly failed.
You seem to be basing your entire argument on the belief that police activity does not deter criminal activity. That is simply untrue. It also presumes that the alternative the US has used, engaging the enemy with primarily military means, somehow is a more effective deterrent, when study after study has shown that it has turned many otherwise politically ambivalent people into combatants.
Further, you state that policing has failed. The United States hasn't even tried that approach.