Very many professions/trades use just one word for the practitioner - without regard to the gender of the practitioner:

Doctor, dentist, lawyer, engineer, counsellor, librarian, nurse...

But the stage gave rise to "actor" and "actress" - why, I wonder? Like "waiter" and "waitress". The world of the arts also has "artist" and "artiste". Hmmm... must try out "dentiste" sometime...

"Barman" and "barmaid" specifically mention the gender - but I must say I prefer those terms to "barperson". And I have the same opinions as Little Hawk, on the change of terms used for those chairing a meeting.

The illogicality of language is part of its fascination.