"Jesus saves us not because he died on the cross, that is just a spectacular example of incomprehensible self-sacrifice."
It is that, but I do think you missed the part where he ROSE from the dead. His sacrifice was impressive and might well have gained some loyalty points, but the reason we are "saved" is because of Jesus' power to conquer death. And the New Testament is full of people saying that Jesus did not, in fact, prevent them from sinning (although they gave it a good try), the point is that Jesus has the power to forgive them *even though* they descend into sin anyway.