A good deal of awkwardness (and "he or she"/"his or her" etc. *is* awkward, while "he/his" though it *legally* and according to the grammar books, refers to both men and women, nevertheless does in real life suggest a man) could be avoided if we simply used the words "they" and "their" as, for example, Caxton, Shakespeare, and Chesterfield did: "Each of them should...make themself ready." (Caxton) "God send everyone their heart's desire." (Shakespeare) "If a person is born of a gloomy temper, they cannot help it." (Chesterfield) According to the article "Language and Gender" in *The Linguistics Encyclopedia* the use of masculine pronouns to include females "only became a general rule in 1746 when John Kirkly [sic--it's Kirkby--DT;see
https://books.google.com/books?id=VaBgAAAAcAAJ&pg=PA117 for the rule in question] made it the twenty-first of eighty-eight grammatical rules, on the ground that the male pronoun was more comprehensive than the female. Later grammarians added to this feeling the notion that the use of *they* violated rules of number agreement--a consideration, which, as we have seen above, did not concern Shakespeare, and one which appears to make the unwarranted assumption that number agreement is more important than gender agreement..."
https://books.google.com/books?id=OZ-HAgAAQBAJ&pg=PA349
H.L. Mencken's comments in *The American Language* are still interesting:
"The American pronoun does not necessarily agree with its noun in number...At the bottom of this error, there is a real difficulty: the lack of a pronoun of the true common gender in English, corresponding to the French *soi* and *son*. *His*, after a noun or pronoun connoting both sexes, often sounds inept, and *his-or-her* is intolerably clumsy. Thus the inaccurate plural is often substituted. The brothers Fowler have discovered 'anybody else who have only *themselves* in view' in Richardson and 'Everybody is discontented with *their* lot' in Disraeli, and Ruskin once wrote, 'If a customer wishes you to injure *their* foot.' I find two examples in a single paragraph of an article by Associate Justice George B. Ethridge of the Supreme Court of Mississippi: 'We should keep it possible for anyone to correct *their* errors' and 'No person can be happy in life if *they*'; and another in a war speech by Woodrow Wilson: 'No man or woman can hesitate to give what *they* have.'..."
https://books.google.com/books?id=cIPoCHsgATUC&pg=PA460
(As Mencken observes, this is the line as Wilson spoke it, according to the newspapers reporting it. It was later "corrected" in the written version.
https://books.google.com/books?id=G-WrlQMn8OYC&pg=PT676)