Is there any documentation of when hair-care professionals' perception changed from mostly male (barbers) to mostly female (hairstylists, as it is today)?

Prior to short hair for women became popular and the invention of chemical perm/chemical hair dye (invented after 1870's but became popular after World War I) there were little need for women to visit a professional. Occasionally there were male hairdressers for women such as Léonard Autié who did Marie Antoinette's hair but most women only needed their maids to help them with their hair.

However by the 1950's there were definitely plenty of female hairstylists. For example as early as 1906 Madame C. J. Walker taught African-American women to become hair stylists and sold her hair products. She even opened a beauty school training black women (here are some of the graduates). As more and more hair styles and products for women that required professionals to use became available in the 20th century (as well as more women are working outside their homes), more women became hair stylists.

Barbers by definition are groomers for men. Not only they offer haircuts but they also shave clients with straight razors. You don't see a lot of male barbers anymore because the concept of barber shops as a place for men to get groomed and socialize have been declining.