"They are basically children of their culture and know no other way." I understand where you're coming from here. It's the ideology we're fighting, not the people. We've cured some rotten ideologies in the past. The point is, they're going to have to abandon their culture. It preaches death.
Our other past successful wars have followed a general pattern:
1) Identify the enemy
2) Attack all infrastructure, supply-lines, and outside sources of support
3) Engage the enemy military
4) Demoralize non-combatants
5) Isolate the command structure
6) Obtain unconditional surrender
7) Help with the rebuilding process
There are some very real psychological reasons that these steps need to be executed in order. By engaging in the 'hearts-and-minds' campaign before the enemy is defeated, we're setting ourselves up for failure.
The tragic part of this is that the ideology that needs to be destroyed is wrapped-up in their faith. For a full surrender to be made, all of their Mullahs will have to denounce Mohammed's rants about killing infidels. Those verses will need to be stricken from the Quran, IMO. The actual shooting may temporarily subside, but until that's done, Islamists will carry on to the eventual aim that we're all either converts, slaves, or dead.
I'm not sure the time-tested style of carrying out a war will work in this case. Fortunately, the most ideological Islamists are the ones most willing to actively engage in jihad and so make themselves available for their just reward. Since these types are scattered amongst the innocents, the administration's fly paper/magnet strategy in Iraq is the preferred method and has actually worked quite well. Instead of waiting patiently for airfares to the US to drop, the theater and subsequent death in battle are readily available to any jihadist willing to take a half day's ride in the back of a truck to Iraq. Then all we have to do is hope the rest of them divorce how they live their lives from what their faith preaches- kinda like a lot of Christians and Catholics these days, unfortunately.
You are correct. Basic Islamic ideology is incompatible with life in a society based on Judeo-Christian principles and accordingly, needs to be reformed. Relying on hope for reform is a bit too passive. Education is key. We do need to reform their education systems to a secular western style and more importantly, preach the gospel vigorously.