You stopped before the part that this thread addresses.
So we have this:
Atheist: You can't prove that god exists!
Theist: Prove that god doesn't exist!
Atheist: You have the burden of proof!
This is the argument you described in your post. But, the debate doesn't just end there. My question is "What about what comes next?"
Theist: God exists because argument X, reason Y, and evidence Z.
(This is the theist's attempt at fulfilling her burden of proof.)
Atheist: You have not convinced me, therefore, you have failed to meet the burden of proof.
(This is the atheist rejecting the proof offered by the theist.)
I am not arguing about the part of the debate you described in your post. I am talking about the next step. The theist has offered proof and the atheist has rejected it.
Does the atheist need any reason to reject the arguments offered by the theists?
Is she allowed to reject them "just because"?