I've worked as a copy editor with numerous in-house style guides at different publishing houses, as well as with various style guides intended for a wider audience (Chicago, AP, MLA, Oxford, Words Into Type, Harvard Blue Book), and I can't recall ever having seen one that imposed an alternating-gender-pronoun approach. I have occasionally encountered this approach in books and periodicals, and have supposed that the author or publisher adopted it to emphasize the randomness of assigning gender to a person in a particular occupation or to a generalized human being, but I've never seen it required as a matter of house style.
The reason that it hasn't caught on as an approach to gender neutrality, I suspect, is that it has the effect of making gender more prominent in the course of a book or article. Whereas "he or she" or "she or he" or "(s)he" or "they" declines to assign a single gender to the hypothetical or representative person in a narrative, and indeed disposes of the issue of gender specificity by assigning both genders or neither gender to the working pronoun, the alternating-gender approach insists that each such pronoun—and the person it points to—is either male or female.
Rather than emphasizing (as I imagine it intends to) the interchangeability of male and female pronouns in generalized or hypothetical settings, it emphasizes that this first person is female, this second person is male, this third person is female, and so on. The reader is presented with an endless series of pronouns with alternating assigned genders instead of dealing with a text where gender assignment is avoided because it is unnecessary and irrelevant.
Undoubtedly, the distracting aspect of systematically alternating the gender of pronouns applied to generic individual people would diminish if the practice became the de facto standard in speech and writing, but even then I don't see how it would offer any meaningful advantage over the gender-neutral alternatives that are currently more common.
Best Answer
Old English was highly inflected and the third person singular personal pronoun had masculine, feminine and neuter forms. The neuter form was, in the nominative, hit, which became modern English it. English grammatical gender has disappeared, but we retain he and she to refer to nouns which clearly describe male and female entities, mostly people. The neuter it refers to everything else.