The use of they with a singular antecedent goes back to Middle English, and [...] it has continued to be very common.
(The Cambridge Grammar of the English Language, ed. Huddleston and Pullum, p. 493)
In fact, this usage ("singular they") is to be found in Shakespeare and the King James Bible. It is as typical for the English language as the use of plural verbs with certain singular nouns like "police".