I don't like the singular they. I find it far more annoying than the generic he or the construct of he or she.
The generic he actually has an interesting history. In Old English, the word wifman (-> woman) is a masculine noun and wif (-> wife) is neuter. This is because man though meaning basically "human" is a masculine noun so any compound of -man is also masculine. If one wants to specify a male man in Old English one either uses wer (counterpart to wif, survives in werewolf, which is why I joke that all werewolves are by definition male) or wapman which is the counterpart of wifman.
This led to an interesting problem. If you are talking about a specific woman and have perfect gender agreement, then "he" has to be used to refer to "woman." That makes things hard to read and so Old English has a rule which is that you ignore the grammatical gender of the word when you refer to a specific person. If we applied this to modern English we'd have amazing constructs like:
"I went to the store and the woman at the cash register told me that I needed to give her an extra $5."
vs
"Any woman thinking about getting married should make sure that he has some level of financial independence first."
The first is specific while the second is generic. He in the second refers to "any woman" not her prospective spouse!
Similarly "My wife says she isn't up to coming to the party this weekend" but "A wife today still finds it has to do a majority of the housework."
I don't like mixing numbers though. If "they" is an acceptable singular why isn't "it's" an acceptable possessive?