I must agree with the answer by Michael Seifert that fair use cases can be complex, and are usually highly fact dependent. They are indeed decided on a case-by-case basis. That said, I would not describe them as "a morass of grey areas". I also differ in my view of a likely fair use analysis in the case in the question, although the statutory factors are listed correctly, if not in my view always explained correctly.
Purpose and character of the use, including whether the use is of a commercial nature or is for nonprofit educational purposes.
That Mr A is not acting from commercial motives probably tilts this factor toward fair use. Moreover an effectively "documentary" work is probably broadly "educational". This factor is usually where analyses of the "transformative" (or not) nature of the use is placed. Note that for a fair use analysis a work is "transformative" not because it is modified in some significant way, but because it is used for a significantly different purpose. For example, when a popular song or poem is published, the intent is to entertain, and to induce specific emotions in the audience. But if these exact same words are quoted in a text on the art of verse, they would be used to show techniques of rhyme meter, and the like, a highly transformative use, in spite of the words being identical. (The official US Copyright office page "More Information on Fair Use" reads: "Transformative uses are those that add something new, with a further purpose or different character, and do not substitute for the original use of the work.") In the case suggested in the question, the purpose would be to support the conclusions in A's book, which is at least somewhat transformative. I conclude that this factor overall would most likely lean toward a finding of fair use.
The Nature of the copyrighted work.
In general a work that is largely factual gets less protection than a work of fiction or verse. A work with much creative narrative recounting facts is somewhere between a dry textbook and a novel. A "landmark article" might well be in that same middle ground. But in any case this is not as often the factor that is decisive in fair use decisions.
The amount and substantiality of the portion used in relation to the copyrighted work as a whole.
This factor probably leans somewhat against fair use, because the whole article is probably being copied. But this factor has been more likely to weigh strongly against a finding of fair use when the amount used is significantly more than is needed for the legitimate purpose for which it is used. There are many cases where the use of an entire work (particularly a relatively short work such as an article or a poem) has been held to be a fair use.
The effect of the use upon the potential market for or value of the copyrighted work.
Significant documented effect on the actual current market for the source work tends to weigh most strongly here in case law, although the statute does not say so. Effect on a currently unexploited market that might be exploited in future is sometime quite significant. But when the copyright holder has intentionally ceased exploitation, in particular because there seemed to be no further market, this is less likely to be a significant factor. The same is true when it appears to the court that there is no viable commercial market to exploit. If the newspaper or magazine has no history of republishing articles after it has dropped them from circulation, that would weigh towards fair use. Also when copies are supplied only on an individual basis, after a personal request, that would be less likely to harm any potential market for the article. If A indicates that copies will no longer be provided to others when and if the article is republished and made available by the copyright holder, that would probably significantly reduce the weight of this factor. (The official US Copyright office page "More Information on Fair Use" reads: "Here, courts review whether, and to what extent, the unlicensed use harms the existing or future market for the copyright owner’s original work. In assessing this factor, courts consider whether the use is hurting the current market for the original work (for example, by displacing sales of the original) and/or whether the use could cause substantial harm if it were to become widespread.") This factor might well be close to neutral, depending on specific facts brought out during an infringement case.
17 USC 107 also says:
The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors.
Note that for purposes of copyright law "unpublished" normally means "never published" not "out-of-print". This is because it is considered that an author has a significant interest in deciding when and if (and how) to first publish a work. It thus takes a stronger than usual case to justify use of a never published work as a fair use (although this has been justified in many cases). But this consideration does not apply to a work that has been published and then allowed to go out of print. For such a work, the bar for fair use is lower because public access is considered a public good and part of the values that the copyright law is intended to foster.
See also the case of Suntrust Bank v. Houghton Mifflin Co., 268 F.3d 1257 (11th Cir. 2001) in which the US Court of Appeals for the Eleventh Circuit held that The Wind Done Gone was a fair use of Gone With the Wind. In that case, th Court wrote:
In assessing whether a use of a copyright is a fair use under the statute, we bear in mind that the examples of possible fair uses given are illustrative rather than exclusive, and that " [a]ll [of the four factors] are to be explored, and the results weighed together in light of the purposes of copyright." Campbell, 510 U.S. at 577-78, 114 S. Ct. at 1170-71.22 ... one of the most important purposes to consider is the free flow of ideas - particularly criticism and commentary.
Despite whatever educational function TWDG may be able to lay claim to, it is undoubtedly a commercial product.24 As the Supreme Court has stated, " [t]he crux of the profit/nonprofit distinction is not whether the sole motive of the use is monetary gain but whether the user stands to profit from exploitation of the copyrighted material without paying the customary price." Harper & Row, 471 U.S. at 562, 105 S. Ct. at 2231. The fact that TWDG was published for profit is the first factor weighing against a finding of fair use. Id., 105 S. Ct. at 2231. However, TWDG's for-profit status is strongly overshadowed and outweighed in view of its highly transformative use of GWTC's copyrighted elements. " [T]he more transformative the new work, the less will be the significance of other factors, like commercialism, that may weigh against a finding of fair use." Campbell, 510 U.S. at 579, 114 S. Ct. at 1171. " [T]he goal of copyright, to promote science and the arts, is generally furthered by the creation of transformative works." Id.. A work's transformative value is of special import in the realm of parody, since a parody's aim is, by nature, to transform an earlier work.
While the assessment of a parody of a novel is significantly different from that of redistributing a newspaper or magazine article, the comments on commercial purpose and transformativeness are, in my view, relevant.
In short, I think the kind of use described in the question is significantly more likely than not to be held to be a fair use under US law, although much would depend on the details not included in this hypothetical.
I should also mention that Mr A has several other options which are less legally questionable.
First, A could ask the author or copyright holder (possibly the publication where the article appeared) for permission to send out copies. If permission is granted, A is fine and the issue of fair use does not arise.
Second, A could simply try to find out if an archive site, such as the Internet Archie, has captured the original online version (there are several such sites). If it has, A can provide a link to the archived version without any need to make his own copies.
Third, the publication may have its own archive -- many do. If so, A can link to that.
Fourth A can simply cite the original with enough detail that it can be found in a good library, and advise readers to do so.
Fifth, A can summarize the original article without copying it directly, providing only enough detail to support his own book, and combining this with one or more of the methods 1-4 above. If done properly (including proper attribution), there should be no copyright issues with such a summary.
None of these require depending on a judgement as to whether distributing a copy is fair use or not.
If A does plan to distribute copies of the original article, A might want to consult a lawyer with copyright experience. The cost for a single consultation might not be too high.