The very first and most obvious problem is the way in which Faust links argument and persuasion:
The primary purpose of an argument, understood in the philosophically orthodox sense, is to persuade someone of the truth of its conclusion. This aim of argumentation is so obviously and widely recognized that it is often written into the very definition of the term. A typical account of argument, found in a standard introductory logic textbook, deﬁnes one as “a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion).” Now, typically we understand the arguer and his audience to be separate persons, such that the statements in an argument offered by person A aim to provide another person, B, with reasons for B to believe the conclusion of A’s argument. On this view (hereafter called “the standard view”), an argument’s primary purpose is to persuade an audience to accept its conclusion.
However, the primary purpose of an argument is not to persuade someone of its truth; the primary purpose of an argument is to relate a claim correctly to reasons indicating it. And we can see this immediately even in the very example Faust uses. The introductory logic textbook (Hurley's), defines an argument as "a group of statements, one or more of which...are claimed to provide support for, or reasons to believe, one of the others". But providing support for, or reasons to believe, is not the same as persuading, and if the aim of an argument is to provide support for a conclusion, this is distinct from persuading. Providing support depends on the facts of the case and the logical structure of the argument. Persuading depends, at least in addition, on the psychology of people presented with the argument. Further, Faust's conflation of the two is inconsistent with how people actually use arguments: many arguments only make their conclusion more probable (Faust herself later recognizes this), and have to be used in conjunction with other arguments, and in such a case, the individual arguments will not be expected to persuade on their own. So even if it were true (it is not) that argumentation generally speaking has persuasion as its primary purpose, this is completely different from saying that an argument has persuasion as its primary purpose.
The most longstanding and historically influential account of how argument relates to persuasion is the classical/neo-classical account based, essentially, on Aristotle, Cicero, and Quintilian. It is inconsistent with the idea that Faust presents here. The classical account recognized that persuasion requires the combination of three elements: ethos or character, movement of the pathoi or passions, and logos or rational account. All three are always operative in persuasion, which is why no one in the classical tradition holds that rhetorical arguments work exactly like demonstrative arguments: rhetorical arguments, arguments for the purpose of persuading, are modifications of the primary kinds of arguments (demonstrative and dialectical) in order to express more clearly the good aspects of the character of the persuader and to move more easily the passions of the audience. Among the most important post-classical accounts, Campbell holds that arguments to persuade have to appeal to all the major faculties of the mind, not just reason, and even Whately, who most thoroughly unites logic and rhetoric, does not hold that persuasion is the aim of argument generally.
Thus the 'standard account' is anything but standard. It's possible, of course, in this age of advertising and political spin, that it's making a comeback, but even Faust's paradigmatic example doesn't actually express the 'standard account'. It should be said that Faust's whole point in the article is to argue against the 'standard account', at least for religious arguments; she herself goes on to recognize that there are arguments that lack persuasive force despite being of sterling soundness, and that the capacity of an argument actually to persuade depends on things external to any argument. And she will later give a whole list of other possible purposes for argument.
Faust's own conditions for persuasiveness of arguments are somewhat interesting. In rough summary, she holds that the necessary conditions for an argument to persuade are that
(1) its premises are 'subjectively probable' for that person;
(2) these premises are recognized by that person as making the conclusion more probable;
(3) the premises are more 'acceptable' to that person than the conclusion, where 'acceptability' has a technical meaning.
As she notes, (3) is controversial if taken as a general requirement. She argues, however, that it holds in dialectical contexts. This can be conceded for the sake of argument, but it's worth pointing out that this is another point on which Faust is inconsistent with the classical tradition, which takes dialectical contexts as precisely those where (3) cannot be assumed. For instance, Aristotle's counterpart to (3), which is actually in terms of knowledge rather than acceptability, only applies to demonstration; in dialectic (which concerns inquiry) and rhetoric (which concerns persuasion) it does not necessarily apply. This is quite clear in dialectic, where it is sometimes necessary to show that better-known (or more acceptable) claims are implied by less-known (or less acceptable) premises, and also in rhetoric, which has so many additional factors (what people can accept in public, the passions, character, etc.) that can interfere with acceptability as understood in (3) without actually affecting an argument's ability to persuade.
This allows Faust to formulate an account of what she calls 'begging the doxastic question'. Unfortunately, she gives two different accounts and treats them as if they were the same. The first account is
An argument begs the doxastic question, on my account, when a subject would ﬁnd the argument persuasive only if she antecedently believes the argument’s conclusion.
The second account is,
If an argument begs the doxastic question, then the assignment of some positive degree of probability to at least one premise relies on acceptance of the argument’s conclusion. But in so fulﬁlling condition (i) on persuasion, the argument violates condition (iii).
These are not equivalent. (1) The persuasiveness of an argument, which is the key feature in the first account, is by Faust's own account not dependent only on the acceptability of the premises, which is the key feature in the second account. (2) Faust herself distinguishes acceptability from belief. (3) The first account applies to all arguments whatsoever, because the persuasiveness of any argument always presupposes that we do not think its conclusion is false. In order to accept an argument we have to have reason to think that it has not become an unintentional reductio, whether to absurdity or definite falsehood, and that the argument is not even put into doubt by the doubtfulness of its conclusion. Conclusions are assessed in terms of reasons, and the combination of reasons and conclusions is an argument, but an argument has to be assessed holistically. This contrasts with the second account, which does not obviously apply to all arguments, but only to arguments insofar as they can be used in contexts in which a certain psychological fact (about about the relation between 'subjective probabilities' of the premises and that of the conclusion) are true.
Since it's quite clear that Faust does not intend to argue that all arguments beg the doxastic question, which the first account requires (since you can't find a whole argument persuasive until you've come to believe the conclusion) we'll set aside the first account and simply focus on the second: fulfilling condition (1) involves violating condition (3). We can take this as simply definitional, whether we accept (3) as a necessary condition or not. Now, the first thing to note is that whether an argument begs the doxastic question in this sense does not depend on anything to do with the argument itself, but on a counterfactual claim about the psychology of possible audiences for the argument: it would require some general evidence about psychology to support. And here we run up against a problem. When Faust argues that religious arguments (she includes both atheistic as well as theistic arguments in the mix) beg the doxastic question, she repeatedly uses the fact that someone who accepts the opposing position will likely reject one of the premises. But if we take it this way, it will create the same problem for the second account that we saw with the first account: It is true of every possible argument, regardless of subject. That is, we can say of any argument that "in certain dialectical contexts in which the audience is assumed to believe not-C prior to the reception of [an] argument concluding that C, such...arguments are unlikely to provide such an audience with reasons to believe that C". This is due to the very same fact that I previously noted, and is confirmed by experience: people who disagree with conclusions don't generally change their minds simply on being presented with arguments for the conclusions, because even with rigorous arguments they look first to see if they can consistently get away with rejecting the premises. Faust's argument, if correct and taken in the way she applies it, would be a reason to disagree with the 'standard account'; but it doesn't say anything about religious arguments in particular. It's not necessary to go as long a way around as Faust in order to reject the 'standard account', however; any serious look at argumentative practice of almost any kind shows that it is false. It is likely, though, that she is still conflating the first and second account, and thus persuasiveness of argument with subjective 'probability' of claims.
It should be noted, however, that Faust's manner of proceeding is faulty, even if we don't understand it in the way she has to for her method. Setting aside the controversial move of taking facts about 'reasons to believe' to be facts about 'subjective probabilities', the mere fact that "in certain dialectical contexts" the argument would not persuade is not strong enough to support the conclusion that the argument can only fulfill condition (1) by violating condition (3). For that to be true, there would have to be no dialectical contexts in which it could fulfill condition (1) without violating condition (3). In order to determine this, for instance, we have to consider not only cases where people already disagree with the conclusion, but also cases where they are open either way, and in addition cases where people are already leaning to the conclusion, although they still regard the premises as more probable than the conclusion. If people are persuaded in any such cases, then the fulfillment of condition (1) does not in those cases violate condition (3), and one cannot argue on Faust's account that the argument begs the doxastic question. Thus these kinds of cases would need to be considered; all of Faust's arguments that given arguments beg the doxastic question are incomplete.
The upshot, in any case, whether you accept Faust's argument or reject it, is that the 'standard account' is untenable: it's absurd to evaluate arguments based on persuasiveness, because (1) this is not the primary use of argument for any rational person, since even accepting the conclusion of an argument requires more than just being presented with the argument (you have to evaluate whether it is better to accept the conclusion or reject the argument, which requires higher-order reasoning and will often depend on many things other than an argument, much less the argument itself); (2) anything can persuade somebody, so it would vary considerably from person to person and from group to group; and (3) almost all major accounts of the relation between argument and persuasion regard arguments as having a role in persuasion but not the only, or even always the definitive, role. The primary value of argument in persuasive contexts, in fact, seems to be not that the argument as such persuades, but that it contributes to persuasion by establishing the cost of not being persuaded. This in itself can persuade, when an argument shows that you have to give up or accept too much in order to reject the conclusion; but this is not written on the face of an argument. The 'standard account', which takes arguments as existing for the purpose of persuading, shows an egregious lack of understanding about what rational arguments are, or how they function in inquiry. Arguments are often preconditions for persuasion (depending on the kind of persuasion); they often can persuade, when taken in their full actual context; however, they do not exist because of it, nor are they always well-suited for it. As far as arguments themselves are concerned, persuasion is irrelevant: they identify what is consistent and inconsistent, and to this extent what is rational and irrational, neither of which depends on whether anyone is persuaded at all. If you are not persuaded by an argument, that is a fact about you, not about the argument. The argument is good or bad, right or wrong, on its own terms, and it is the argument that shows whether and why you are right (or wrong) to be persuaded -- your persuadability doesn't say anything on its own about whether the argument is good or bad, right or wrong. Your being unpersuaded could be because the argument is bad, or it could be because you are being irrational, stubborn, or perverse, or it could be because the argument, despite being right, conflicts with too many of your prior commitments, or it could be because I have presented the argument badly and in a way that your initial impulse would be against it no matter the argument. We simply don't know without further investigation.
And, what is more important, this is the rational way of handling arguments. It would be rationally absurd to accept a conclusion just because an argument was given for it, no matter how good the argument; at the very least you always have to ask whether it is better to accept the conclusion or reject the argument, and why. Rational people accept arguments not on their own but as they fit into the broader context of our rational lives, one in which all the elements noted by the classical tradition -- ethos and pathos and logos -- are always operative. It is not possible to turn any of them off; if anyone thinks that any one of them was not operative in any given case of persuasion, he is only deceiving himself. Rationally being persuaded involves the harmony of these elements. But arguments do not exist for persuasion; they are relevant to persuasion because of the way our minds work, and for no other reason. Arguments can be for nothing more than showing that reasons can be given, or what kinds of reasons might be given, for instance; or they could be for connecting different fields of thought; or they could be just for showing what's consistent and what's not. None of these are persuasion, although any of them can be relevant to persuasion.