"I am asking how you and/or science are qualified to make such an assertion. If science is not competent to make statements regarding God, then how can it with competence state that God is outside of its purview?"
Because, as YOU have already said, there is no way to make a weighted choice between the statements *Everything can be explainable by God* and *Everything can be explainable without God*. If you can't make a weighted choice, you can't say one is a better choice than the other. You are stuck. That is where science is; it's stuck because there simply is no way to decide if God does or doesn't exist. As YOU have already agreed.
"You misrepresent, or misunderstand what I've said."
No, you misrepresent and misunderstand what YOU have said.
"I accept two types of science as having validity, neither of which operate under falsifiable assumptions. Furthermore I made clear that it is incumbent upon every observer to make a choice based upon those two, and gave the options."
But you said there was NO WAY TO CHOOSE between the two that wasn't better than the other. The proper stance is then agnosticism regarding this question. Theism and atheism are never scientific stances. Not yet anyway; if more info is produced to allow one to make a weighted choice, this may change.
The choice resides with each observer. Some make their choice blindly and do not allow their point of view to be challenged while insisting their point of view is the only one supported by the United States Constitution. Others weigh the evidence as it comes at them and then choose which general understanding is more explanatory. Some people are even capable of seeing things from more than one point of view. Can you believe it?! Since neither approach to science is falsifiable, we'll have to make do with either approach or some admixture of the two. Just as there is more than one way to interpret and explain the motions of the planets, there is more than one way to engage science.