No, it doesn’t “expose yet another bias”.

There has to be one single solution to the riddle. Those are the societal rules of how riddles work. You can’t just say, “The answer? The doctor could have been anyone!” and not expect your audience to roll their eyes and not want to listen to you next time.

Allow the doctor to be a woman, don’t insist that the doctor just has to be a man, and let it go.