In old English, males were called "werman" and females "wyfman". "Man" was gender neutral, and has been in use in our language to describe people, in general, since that time.
Despite criticism, it is not incorrect to use "mankind" in writing.
Mr. Barzun has written "that the word 'man' has its roots in the Sanskrit 'manu,' which means human being," you note, continuing: "Its counterpart in Latin is 'homo.' Thus, he argues, the word 'man' has always had a double meaning -- both male human being and human being in general. . . .
It is true that the word "man" -- which appears to derive from the same Indo-European root as "mind" and thus originally meant thinking creature -- has long had the two meanings mentioned, both of which are attested for Sanskrit "manu."