I'm not disagreeing with you on this specific point, only on the last one.
I realize that people would still be killing each other even without religion. There is, however, a very important distinction to make here. The people actually blowing themselves up are motivated by religion. What you're referring to (people killing each other regardless of faith) more accurately describes the guys sending the suicide bombers to do the killing. In other words, the guys with knowledge, influence and power are the ones who brainwash/delude suicide bombers into suicide bombing. They use religion as a justification because it's convenient; even if religion were absent, they would find another justification (such as politics) to send their able and willing troops to their deaths.
However, this does not change the fact that the guys actually blowing themselves up (the suicide bombers themselves) are motivated almost exclusively by religious belief.
