(1) I'm a little puzzled at Hajek's puzzlement over Hume's apparent qualification of his argument:
I beg the limitations here made may be remarked, when I say, that a miracle can never be proved, so as to be the foundation of a system of religion. For I own, that otherwise, there may possibly be miracles, or violations of the usual course of nature, of such a kind as to admit of proof from human testimony....(SBN 99 (p. 127)
Hume seems to explain himself pretty well, I think, since the claim is summarizing the argument in the previous paragraph, which in turn is summarizing the argument as a whole. There he gives a quick overview of his balancing view of contrary probabilities, notes that on his arguments "all popular religions" are balanced against such probabilities that there is a total "annihilation" even at best assessment. Thus, he concludes, "no human testimony can have such a force as to prove a miracle, and make it a just foundation for any such system of religion." Or in other words: because of psychological issues that arise in the case of popular religions, there is no testimony that can give such grounding to a miracle that we can base a religion on it.
After making the above qualification, he gives examples to clarify his point, by considering three hypothetical scenarios.
(1) The Eight Days of Total Darkness. In this case, the testimony is assumed to be unanimous, or nearly so, and the event is supposed to be analogous to things that happen in nature. Thus it almost comes within reach of human testimony on its own; with solid testimony one might well believe it.
(2) An Apparent Resurrection of Queen Elizabeth. In this case, the testimony is universal among historians, who record a curious confluence of circumstances suggesting that Elizabeth died and then was later alive. Such a strong evidence for such an event would be very surprising, says Hume, but even very surprising things are more plausible than such a radical violation of nature.
(3) This miracle ascribed to a system of religion. The introduction of religion introduces such a mix of knavery and gullibility that we can reject it out of hand.(I'm sure you catch the hint here, of an apparent resurrection ascribed to a system of religion.)
What this all means is fairly simple. Hume's balancing account does not of itself rule out accepting the occurrence of a miraculous event; it merely sets the bar for accepting one very, very high. And in particular, the psychological conditions have to be just right, so that we can accept that the people giving the testimony were not foolish, and were not liars, and are testifying to something that is not utterly incredible. Hume rejects religion-founding miracles because the psychological conditions in such cases are not just right and on his account cannot be. There is nothing mystifying about Hume's concession to the hypothetical eight days of darkness. Hajek makes a distinction between extraordinary events and miracles that is superficially similar to the distinction Hume makes between them; but only superficially so. What distinguishes the two for Hume is that we can't say that events violate the laws of nature if we are ignorant of the circumstances in which the events occur; an extraordinary event differs from an ordinary event by occurring under very different circumstances, whereas a miracle differs from an ordinary event by being very different under the same circumstances. Campbell and some others argue that this is actually giving away the whole store; but whether it is so or not, it is clear that the eight days of darkness occur under normal circumstances. Indeed, they occur under circumstances as normal and ordinary as one could wish, the astronomical facts of the earth's revolution and rotation and the meteorological facts of the world's standard weather patterns. To file the eight days under the label 'extraordinary event' Hajek would have to show that they did not occur under the ordinary circumstances. Since it's a purely hypothetical example, and Hume himself doesn't suggest otherwise, this is impossible.
(2) Hajek thinks that Hume's concession of arguments for contrary propositions, each of which leaves no room for doubt or argumentation, is a slip. I think this is fatal to his interpretation. As Fogelin has pointed, there is nothing inconceivable about having two arguments of completely different kinds, neither of whose general forms you can reject, concluding to propositions that can't both be true from starting points that can't be rejected. It's a tragic situation, but it's possible. And even if this weren't true, it is very, very clear from what Hume says elsewhere that on his account this not only is possible but happens all the time. It's his repeated point in Part IV of Book I of the Treatise. What is more, it is clear that Hume is not committed to the claim that all arguments leaving no room for doubt and opposition are equal; because he is not committed to the claim that they are all equally vivid or forceful. (Indeed, many things he says require us to say he is committed to the claim that they are not all equally vivid or forceful, even if they are all genuinely indubitable.)
(3) From what I've said so far you can probably guess that I think Hajek's premise 6 is rejected by Hume; he doesn't merely say that uniform experience is a direct and full proof against a miracle; he says it can't be destroyed (so as to render the miracle credible) except by a superior opposing proof. And the reason for this is not hard to find. In Hume the force of a proof is not something intrinsic to it; the force of the proof is the psychological force with which it strikes on the mind. What he envisions in Part I of the essay on miracles is a scenario in which two proofs, each of which is forceful enough that we can't reject it, oppose each other. They each make the other incredible (in a straightforward psychological sense), while at the same time making a psychological demand on us, one that cannot be rejected, that we believe its conclusion. But it doesn't follow that they each make the other equally incredible, or that they each serve to make their own conclusion equally credible. Since on Hume's balancing account in every case of opposing evidences we subtract the force of the weaker from the force of the stronger to get the resultant force of belief in the stronger, so we, unable to reject either argument on its merits, can nonetheless be compelled by sheer psychological force to believe one over the other, if one strikes the mind more strongly than the other. Our own belief will be recognized by us as absurd, given an argument that we cannot reject; but the fact will remain: we believe it (although under the recognized qualification that we have to, for other reasons, regard it as an absurd belief, probably the result of our psychological makeup more than any absolute rational foundations). Hume is, in fact, very clear about this point; in a note to Blair on Campbell's criticism of him at this point, he says that the kind of argument against a miracle is a full proof; but that there are weaker and stronger instances of this kind of proof.
Thus Hume's argument in the essay has the following general form:
Part I: Even given that the testimony for a miracle amounts to a full proof, it is counterbalanced by another full proof, the full proof we all undoubtedly have of the uniformity of nature; as a matter of inevitable psychology, if the two proofs are opposed to each other, we could only believe the proof for the miracle if it overbalanced the proof for the uniformity of nature in terms of psychological force.
Part II: Due to psychological issues associated with religions, no testimony for a miracle in religious cases can amount to a full proof. (But Hume explicitly denies that he is committed to saying that no testimony for a miracle can amount to a full proof in other cases, if the psychological issues don't arise in those cases.)
There are many things that may be said against both parts; but it must be admitted that it's a clean and straightforward argument. It does yield exactly the conclusion Hume claims it does. There is no slip in Hume's concession; it is a deliberate and strategic move in one of Hume's best-designed arguments.
So Hajek's basic claim in refuting Hume's argument is wrong. Hume does not require the probability for a miracle to be vanishingly small; he does not assume this; in fact, he explicitly denies that it has to be.
Nonetheless much of what Hajek says in the paper is still of interest in considering the argument. He does somewhat misleadingly make it sound as if Butler were giving an argument against Hume -- which would, it must be pointed out, be attributing a miracle to Butler, namely, prophecy. Butler, of course, is not considering anything like Hume's balancing principle. The criticism of 'Butler-style counterexamples' doesn't work against Campbell's 'Butler-style counterexample' because Campbell's isn't supposed to be a counterexample to balancing but to something it presupposes, namely, that we can reduce all classes of evidence to the same class. Hajek's defense of Hume's balancing principle simply assumes this reduction.