Should someone having pedophilic urges be considered automatically unsaveable and be killed the second it is found outbin any way?

Our current legal system doesn't kill them on sight, but it might as well. The only thing this level of harshness does is make them get really good at hiding it, and the only sensible and evidence-based way to deal with pedophiles is to create fully anonymous mental health resources that they can access which include behavioral therapy and chemical castration for the most severe cases.