Eliminating the object pronoun "him"

In the last example of the previous post, notice that I wrote "she or he" instead of "he or she." Using these expressions interchangeably is good, because it avoids the subliminal suggestion in always putting he first. But it has a drawback: we're so used to reading he or she that she or he seems unidiomatic and calls attention to the writing. But only the first time and only for the second it