This is just too hard.

"He" is traditional, but often bagged by feminists.

"S/he" is ugly and hardly acceptable in formal or literary contexts.

Sometimes people use "they", even in the singular--even Oscar Wilde did this. But it could be argued that it is ungrammatical.

"He or she" is longwinded and often a rather obvious attempt to avoid offending anyone, except where it is used by a translator being paid by the target word.

Alternating is a real pain, as others have pointed out.

Your suggestion is cute, but has one drawback: it takes away the status of English as a language in which is not required to reveal one's sex.

The author of "A Textbook of Translation", whose name has escaped me, has a really annoying habit of using "she" all the time, which seems completely forced, given that his first name is Peter.

Of course, in many cases you can rephrase to avoid using pronouns or to use only plural pronouns, but this can seem very forced and can sometimes lead to ambiguities or be positively misleading.