Wouldn't that just increase the problem that the person you are responding to brought up? If you capture their facial movements, it might be even easier to tell if they are a man or a woman, despite any scrambling you do.
I feel like this is kinda like choosing who to draft in the NBA solely on free throw percentage.
It would be even more fascinating if you could cluster facial expressions or motions as "more female" or "more male". And for bonus points, work out a way to "filter" or "smooth" the facial signal to reduce/remove these giveaway hints (or insert false hints).