I’m broadly in favour of “they” as a replacement for “he or she”. Yes, I’m not thrilled about the frequently discordant semantics, but we desperately need an epicene pronoun and this is by far the likeliest candidate. Well-meaning efforts over the years to popularise hisher, heesh, hizzer, shehe, h’se, tey, shem and all the others have got nowhere (because, as may have been observed before, not even the most eminent language-improvers were ever in a position to “impose” an idea on an unwilling populace). So now we have a more-or-less functional alternative that has emerged, in impeccably descriptivist fashion, out of common usage.
It’s not quite perfect in all situations, though. When you’re using a plural-indeterminate pronoun to stand for a singular subject, for instance, following up with “they” can force the subject, willy-nilly, to be taken as plural.
For example, imagine you are concealing the identity of a source in a story, and it is important to make clear that your information has come from one whistleblower, but no more than that (because the whistleblower is anxious to protect colleagues from a multiple-suspect witch-hunt, say). If you were to write “I can’t reveal who wrote to us, but they were proved to be absolutely correct”, there is a clear implication from “they” that more than one author was involved – with possibly unpleasant consequences for the likeliest suspect’s close colleagues and associates. If you were attempting to conceal the identity of a single author without being more misleading than necessary, you’d be more or less forced back into using “he or she”.