I think that you've totally missed the point of existentialism if you think that it is merely an argument for atheism that you seem to be misappropriating it as being. Simply put existentialism - even in an atheistic form since it's entirely possible to be a existentialist and still be theist - doesn’t “wear itself out” arguing whether God exists or not, because it sees such arguments as being pointless. I strongly recommend that you read Existentialism is Humanism by Jean-Paul Sartre you sound like you're a bit muddled as to what it is about.
Anyway to quote...
"Atheistic existentialism, of which I am a representative, declares with greater consistency that if God does not exist there is at least one being whose existence comes before its essence, a being which exists before it can be defined by any conception of it. That being is man or, as Heidegger has it, the human reality. What do we mean by saying that existence precedes essence? We mean that man first of all exists, encounters himself, surges up in the world – and defines himself afterwards. If man as the existentialist sees him is not definable, it is because to begin with he is nothing. He will not be anything until later, and then he will be what he makes of himself. Thus, there is no human nature, because there is no God to have a conception of it. Man simply is. Not that he is simply what he conceives himself to be, but he is what he wills, and as he conceives himself after already existing – as he wills to be after that leap towards existence."
"[Existentialsim] declares that even if God did exist, that would change nothing. There you’ve got our point of view. Not that we believe that God exists, but we think that the problem of His existence is not the issue."
Yeah, the argument you're presenting is a red herring.