their, them, they: for singular he, she, his or her.
It was rather like a jigsaw puzzle in which everyone contributed their own little bit of knowledgeAgatha Christie, The Murder of Roger Ackroyd
An error commonly found in both speech and writing, and arising from our lack of a relative pronoun meaning his- or - her.
I wore the fillet of the Golden Cobra, which could only be worn by one who had overthrown the cobra of the seventh ordeal, and in so doing had added strength to their will.(Joan Grant, Winged Pharaoh
In such cases the right pronoun is 'his', unless a woman is clearly referred to.
Eric, old chum, I heartily concur, up to a point. Time has move on and we now have the politically correct habit of using 'she' for the unmarked pronoun, in effect destroying a useful distinction. It is of course an American academical barbarism which by the pervasive colonisation of the banal has come to infect good English. The 'their'and 'they' that you deprecate is at least idiomatic and has the sense of an indefinite ungendered multitude.
Wikipedia has an informative article on gender her specific pronouns.