So the spaces would feel less safe, which is already a failure to serve women's needs.
As I have discussed this topic for the last decade or so, this realization has come to me and gained more and more importance the more I discuss it.
Why is bungee jumping popular?
Bear with me. It's relevant. Ok. It's an analogy, but give me some latitude.
Fear of heights, and especially fear of falling, is an instinct. Fear of falling is universal. It isn't socialized. It isn't something that happened because someone warned you to be afraid of it. Bungee jumping allows people to face their fears, and overcome them. The rational mind can look and see that the operators have taken precaution after precaution. Especially in America, they can be sued for accidents. Your rational mind tells you there is no reason for fear, while your instinct tells you that this is completely nuts and you are going to die. Going against that instinct creates an awesome feeling. "Surviving" the jump also is a thrill, because your irrational mind is telling you that you were in great danger, so the emotional impact is still as if you you had a near brush with death.
I have come to believe that the anxiety associated with being naked in the presence of the opposite sex is not a rational fear in every circumstance. It is not always a logical reaction to the actual possibility of harm. I believe it, too, is instinctive. Some people will say that's ridiculous, because how could we "evolve" as an animal species an instinct to wear clothes when we are not mating. Well, our ancestors started wearing clothes about 5,000 generations ago. That's plenty of opportunity for behavioral evolution.
And if not specifically wearing clothes, certainly humans have sexual signals and the idea of a sexual signal is innate. Perhaps nudity as a sexual signal is not instinctive, but the existence of a sexual signal is instinctive. Nudity is a socially created thing that triggers the sexual signal. So, being naked around the opposite sex, or having the opposite sex be naked around you, is something that can easily trigger a sexual signal. Moreover, I think that human females' fear of sexual assault is instinctive, so that the presence of a male who is making sexual signals, or making sexual signals in the presence of a male, in a situation where the female is not desiring sex, would naturally trigger a fear of sexual assault.
Perhaps there is some element, though, that makes the female's rational mind aware that there is no real danger. Then, it is possible to overcome her (or his) anxiety. (I don't want to misgender any females. This could apply to transmen as well as cisgender women.) Like when bungee jumping, a human female could, in situations where there is no real threat, go beyond the irrational fear of being in the presence of a naked male if a transwoman is using the same locker room.
So, I think that the fear is instinctive, but like other instinctive fears, humans have the capacity to put them aside. So, it is natural for females to fear being undressed in the presence of males, but it can be overcome through rational thought, just as someone can overcome the fear of falling while bungee jumping.
So, should we allow male transgenders (i.e. transwomen) to use the locker room that is primarily used by females? It would seem to make sense, except for one thing:
Women shouldn't have to feel like they're bungee jumping just to go to Spin Class!