The problem with any attempted metaphysical demonstration is the supposedly self-evident principle it invokes. An instance of such a supposedly self-evident principle would be the principle of sufficient reason (PSR), the claim that nothing exists unless there is a sufficient reason for its existence.

The PSR isn't usually said to be self-evident, it's just the basic idea behind explaining reality. In no other context are we content to say, well it just is and think this is a sufficient answer to the question. It may be that we want to put this forward as a reason to reject the idea of any sufficient answer being provided, but again this undermines the entire enterprise of rationality being capable of explaining reality.

Defenders of PSR will point to its intuitive nature and the fact that, in neither our scientific or mundane lives are we content with the “explanation” of the old bumper sticker, “shit happens.” We always want to know why it happens. However, if we deny the existence of brute facts, then we either have to say that the chain of causes that we invoke to account for any phenomenon either extends ad infinitum, or it ends with something that has no further cause or explanation, in short, a brute fact. If God is the end of our explanatory chain, then God is a brute fact.

But this ignores the point at the heart of the cosmological arguments, and the very reason for positing God as an adequate brute fact, that God is a particular entity that isn't supposed to be contingent.

As I see it, naturalism, or—as I prefer to call it—physicalism says that the fundamental features and (if the universe is conceived as having a beginning) the primordial state of the universe are to be taken as given. That is, the universe, or rather its set of basic and original features, is taken as a brute fact, neither having nor needing any further explanation.

This is very convenient, but it gives no answer to why what appears to be a contingent entity (the universe) should be taken as a given. It also gives no answer as to how a universe that did begin to exist could have come from nothing.

With respect to the question “Why is there something instead of nothing?” the full, complete, and satisfactory answer is “Why not?”

But the theist has proposed a reason "why not" in the various forms of cosmological argument, so this isn't a complete answer, and it's not a satisfactory answer since it ignores the reason the theist has given.

All arguments for the existence of God ultimately rest upon metaphysical intuitions, However, it seems to be the nature of all such intuitions that they admit of rational doubt.

I'm not sure how this could serve as a critique of theistic arguments in particular, since it applies to all metaphysical models. I'm also not sure why our intuitions admitting of rational doubt is a problem for any of the proposed metaphysical models unless we're going to insist on absolute proof, rather than accepting the best available explanation given the evidence we have available.