" yet science seeks to explain ALL aspects under God's purview, including the human soul."
No, science has nothing for or against to say about the existence of a soul.
" There is NO field where science does not say: I can explain that, God; you are not needed here."
You are mistaking the epistemological reality that science has limits (the existence of God being outside it's scope) with the claim that science says there is no God.
Read some papers by social Darwinists.