Explanatory strength is IMO irrelevant -- as I said, you can have an internally consistent system that explains reality to the tiniest minutia and it still can be wrong. If something can be demonstrated however, it's a totally different matter. Hence, burden of proof.

I also don't think that defining natural as physical is fatal to naturalism. After all, information is energy is matter. No need for anything supernatural here. Neat and simple. But sure, absence of supernatural is fatal for theism.