On the contrary, history has shown that physical destruction is only method that work on ideology.
And since you have asked, Islamic terrorists originated from Islamic dogma which advocates killing of all infidels. If you believe otherwise, go and debate Islam with IS on forums. Extremist muslims own tolerant muslims in debates with such a consistency that i believe that radical Islam is true Islam. An IS mujahid would blow so many holes in RoP narrative, solely using Quran and Shahi Hadits, that " Taqqiya and Kitman " narrative would have more leaks then Iraqi Navy. Don't believe that! Try arguing with a mujahid.
IS terrorists derive power from individuals brainwashed since they were 4 years old, and petro-dollars that Arab sheikhs pump into terrorist machines.
Your narrative of west should act better has a glaring fault. It overlooks the fact that West does not need to prove itself to be better. It could simply close down it's border which most of the countries have done. US has constituted a multi-layer Animal farm program to decide eligibility of visas. Australia has stopped taking in boat people. UK has made it impossible for anyone except super-talented or super-rich to migrate. India has elected a right wing party. UKIP and BNP won EU polls in UK, Le-pen is gaining popularity with each passing day. Germany, already quite conservative, is becoming even more conservative.
Islamic terrorism was the action. What is happening today all over the world is a reaction.
You seem to be reading from the ISIS and AQ handbook, maybe you've got to read on the history, the AQ and ISIS form of violence only appeared after the soviet war and the fathers of the ISIS ideology where pretty much heroes in the west because they fought the "evil" communists, recent history shows that no muslim nation has gone to war against "infidels" solely based on the "infidels" rhetoric so stop this "advocating killing of all infidels", you're dealing with militants not nations, and to defeat them you've got to treat them as such.
Most haven't been brainwashed "since they were 4", most were brainwashed in prisons (most ISIS leaders and expats were in prisons in Iraq or in Europe), and those that were brainwashed, where brainwashed because they were at war with themselves, they were looking for an identity, they were angry young people, to cure this diseases, you've got to get into their heads, most former extremists have identified the problem, it's deeper than just going to a mosque and getting brainwashed.
The west can do better, the muslim world has to do better than the west of course, but the ISIS crisis has been hijacked by politicians with dubious intentions, Turkey, KSA, Qatar etc. with intentions to overthrow Assad, Australia, UK etc. with intentions to change laws due to failed domestic policies (mainly economical and off the scale immigration), there is just so much one can blame "Islamic terrorism" for, one can blame them for the violence in Iraq and Syria and insecurity in the middle eastern and north african regions, but you can't blame them for domestic problems in far away lands in Europe and Australia, let's be realistic, because being realistic is the only way ISIS and company can be defeated.