Ultimately I think the question of whether God exists is irrelevant to religion.
God = a creator, an infinite being, a first cause, etc. None of these understandings of God say anything about religion.
It is possible that God exist? I have no idea, maybe he does maybe he doesn't. However let's say he exists. Let's say one accepts proofs of a first cause or something to create ex nihilo. What have you gained? Nothing about these proofs tells you that God has an interest in men or in giving a strict law code. None of these proofs tell us that God cares about men and most importantly none of these proofs have anything to do with Judaism. You can firmly believe in God but deny Judaism.
Ultimately the only important proof in Judaism is that God revealed himself at some point. Certain thinkers understood this, notably the Kuzari who doesn't base Judaism on a philosophical proof of God but rather on a "historical proof" of the "Kuzari proof".
So why do kiruv agencies and speakers etc. bother "proving God"? Why is it important to them?
I think this is just evidence of the way many of these proofs are formulated. What I mean is that proofs of God are generally made as post-hoc justifications of one's religious beliefs. People believe something and then try to look back at their rather irrational beliefs and try justifying it.
However since these people firmly believe anyways its not so important that the proof exactly match up with the belief.
As long as rational activity about one aspect of the religion can be demonstrated a believer can reflect upon his/her religion and say "hey this religion isn't irrational it's rather clever and it can be proven".
What's important to the believer is the activity of making rational justification not the actual thing proven. Because ultimately the believer doesn't need the proof. The proof is just a way of showing that religion in general is rational. And once God can be proven rationally we make a la pligi and say that all of religion is rational even if we can't quite show how....
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