It’s been used for a very long time, long before all the ‘pronoun’ stuff. He or she is ‘clunkier’ in speech as well as in writing. (You probably used the singular they yourself in the past. Before the new pronoun business, nobody thought anything of it.
I don’t think it’s the habit of ‘authorities’ to release any information at all about someone’s medical issues.
(Generally in a case like this, people with whom the individual would have come into contact are alerted that they may have been at risk.)
I don’t want to belabor something trivial, but, sure, the singular “they” has been around for a long time — mostly in informal speech, but every so often in writing by well-known authors. (In classical literature, this is often a situation where “they” refers back to a singular noun used to represent a large class of individuals, which is different from its use to avoid saying “he” or “she.”)
I still see no reason that authorities can’t release the sex of somebody who has caught a disease; that’s not going to lead to any privacy violation. More generally, I think the exteme interpretation of HIPAA has led to excessive secrecy, keeping the public unaware of medical info that would be helpful if generally known.
As for “they” vs. “he or she” in actual usage, I can’t say that I find “he or she” clunky if used occasionally. When I find myself writing something that requires many uses of either singular “they” or “he or she,” I generally rewrite the passage to avoid the issue.
Even though the singular “they” does have a long history (mostly in informal spoken language), the Wikipedia article on it is chock full of references to non-binary individuals and the like — that’s how it’s often viewed these days, and I don’t want to cooperate in that.