>The closest we got to gender having any significance in English (the form we attempt to speak in the United States) is in the early 1970’s when it was announced that it was improper to refer to the word “their” when male and female were concerned within the same subject. Now we use “his/her”. Progress!

You're missing the point here. The object to "their" isn't for mixed-sex groups, but for the singular:

Wrong: The player brings their bat.
Right: The player brings his or her bat.
Right: The players bring their bats.

Tamar