Even the venerable Fowler's Modern English Usage admits (Page 404) that "in a perfect language, there would exist pronouns and possessives that were of as doubtful gender as 'they' and yet were . . . singular," adding (Page 635) that "in colloquial usage, the inconvenience of having no common-sex personal pronoun, in the singular, has proved stronger than respect for the grammarians, and the one that is available in the plural is made to serve for the singular too."
In his day, to be sure, H. W. Fowler was reluctant to accept the colloquial usage of "their" for "his-or-her." Yet I cannot help but think that he would be persuaded to accept it today, in recognition that pedantic ears are of less importance than personal "ours."