The word ‘man’ appears in two forms in the English language, as a noun and as a verb:
Centuries ago when our King held court, only the King sat, the courtiers were required to stand. From this came the understanding that whoever manned the chair, ran the meeting. The name of the person manning the chair understandingly became ‘chairman’.
Our English word ‘Chairman’ quite obviously means chair handler, and is in no way gender specific.
[similarly, when the cries went out “Man the guns!” or “Man the pumps!” they meant “Handle the guns!” or “Handle the pumps!”]
In conclusion, Chairman is very definitely not gender specific. Pretending it is, does not reveal gender-neutral politics, it merely reveals a sad lack of understanding of the rich heritage of our English language.