You are right this is a cultural struggle. This is a struggle between Shiites and Sunnis. The Sunnis were able to bribe/lobby our congress/politicians to support them so they wouldn't have to fight.
Carrot and stick - I agree, but when we have captured the bad guys we don't let them go. We shoot them just like we did the werewolves of post WWII. 12 year old kid caught shooting a pro allied, German mayor, we shot him! Now we catch and release terrorists and they are hired by Reuters and AP to do anti-American stories about pissing on Korans and such. It’s rather simple you catch a bad guy you shoot them.
A little back ground on me. I've lived half of my life overseas, served as an 11B, and have a PhD along with several grad degrees, speak several languages and have a great time.
Yes I'm extreme and radical sometimes, but when you are dealing with extremists you have to fight them on their own terms. Failing to do that means we will waste of lot of money and lives while the threat grows. And when you put the military in a situation where they are not allowed to fire back at terrorists you may as well pull them out and let the locals handle the problem.
Radical branches of Islam often rise up and the only way to deal with them is to destroy them. And as we are already there we may as well win the war or we better pull out. If we fail to do either we'll create more problems by allowing radical branches of Islam to fester and gain strength. Just like the Sadr’s army, Hizballah, Taliban and Hamas IF the US is going the have a presence in the region we'd better destroy them or get out of the way and let the moderate Sunnis deal with them. Currently our policy of protecting them is only prolonging the struggle and leading to destabilize the region. And yes by tying Israel’s’ hand and not allowing them to have a victory we are only helping terrorists.
By the way, Force is just another branch of diplomacy. One that is used more often as it is illegal for the US to use the other parts of diplomacy; assassinations and bribing. Also I would suggest you read up on the radical Islamists of the past. Learn how other nations dealt with them as well as what gave rise to Shiism. I’ll give you a hint it was the same force that destroyed a radical Islamic band that tried to assassinate the Chingiz Khan (Genghis Khan) and then the established the il’khanids.