I am uncomfortable with the analogy to Jim Crow (as well as racial segregation in countries outside the US). I don't think it applies here.

White people legally kept Black people out of "whites-only" bathrooms with the threat of jail or extrajudicial violence. Black people were not in charge of this system, and white people did not face the same risks if they used Black bathrooms. So, while "women's" and "men's" bathrooms are on equal footing, "white" and "colored" bathrooms were not.

Separately gendered bathrooms may serve a real, benign purpose: giving women a safe place, and enabling modesty for women and men alike. (I'm agnostic about that, at least for purposes of this comment.) But no one today argues, and certainly no one should argue, that racial segregation serves a similar purpose. That is a thing about which we should not be agnostic. We should know the answer.

Generally, no one stands in front of any door and helps you decide whether to open it. So, considering racially segregated bathrooms and gendered bathrooms, there is a superficial similarity—I do emphasize "superficial"—in that people decide for themselves which restroom to enter. But with gender, your decision might be about feeling "comfortable" with yourself, and with race, it's much more about negotiating life amidst violent state power, I believe. I understand that some biracial or multiracial people did pass for white and live their entire lives as white, and I imagine that bathroom choice might occasionally have been fraught for them, especially if they worried they weren't passing as white in a particular moment. But that's different than deciding what bathroom affirms their private sense of their own racial identity. Bathrooms shouldn't be racialized in the first place. It should affirm everyone's sense of justice and dignity that no one has to choose between segregated bathrooms at all.