The gendered bathrooms was actually created to address sex inequality in workplaces. As women entered the workforce in he early 1900's they found they often had only one restroom for the work area, and most would refuse to use it. They pushed for the government to act, and that's why even in workplaces with 99% men there are still an equal number of women's facilities and stalls.
Going to a unisex bathroom would probably upset some transgender advocates as well. Bathrooms are one area of society where we still have and accept segregation. Once all the differences are gone - and they are falling faster than many realize - then there won't be any real difference. Men and women, cis or trans, wouldn't be distinguished except through secondary sexual characteristics, and some would mute them, emphasize them, or simulate them whether regardless of their gender or gender identity.
Right now the fact that they want to use the segregated facility belies the acceptance of and knowledge that men and women are distinctly different and deserving of separate spaces and treatment. Presenting as the gender opposite your biological sex only has meaning while society has cues and differences that are widely adhered to.
Take that away and being transgender loses significant meaning - at least it will for some.
There are transgender people who do not want to change to a unisex society, and will fight such changes as much as social conservatives.