Starbucks already does this and the women hate it. That's because men go in there and leave the seat up, pee all around the rim, and generally make slobs of themselves. Then a woman has to go in there and tidy up before she can do anything. But since it is only women (or effeminate men) that are driving this silly issue, serves them right.
I an assure you I do not go into the restroom and make a slob of myself. I am generally a slob BEFORE I go in.
That would be a solution if this were just about bathrooms. But it's also about gender integration of dressing rooms, showers, girls athletic teams, father-daughter events, basic forms of identification, and anything else you can think of. And it's about abolishing any trace of sexual modesty, since privacy rights have been abolished. The sexual revolutionaries want middle and high school boys and girls showering together, because they favor the early sexualization of children and a culture of casual and early promiscuity. The transgender kids are just the wedge; they are being used to establish the principle and break any institutions that dare to resist.
This is not hyperbole. The revolutionaries take Marcuse as a canonical text. What is being implemented now was avant garde theory just a generation ago. Communists were traditionally casual about rampant promiscuity, and the New Left made it a deliberate, self-conscious ideological norm. The people driving the gender bending regulatory assault are fully conscious of what they are doing.
We saw this commonly at restaurants in Kiev. There’s one door to an open area with sinks that can by anyone and separate doors there for individual toilets. Often there’s bathrooms with a toilet / sink and no gender indicated.