As promised, I will address arguments from Archie (yes I know that's your avatar's name, not yours, but....). After I made that promise, though, Archie said that I shouldn't deal with the whole definition of "man" and "woman" thing. I thought about just dropping it then, because that is indeed the center of controversy, but I hate ducking questions. I will try to address the points only using those terms in conjunction with "cis" or "trans".
I don't particularly like the "cis" words. Indeed, I think they are a part of the gaslighting to which Rolfe refers, but at least they have definitions that we can all agree on. When referring to "trans" people, that will include people that have not yet begun transitioning, or who have no intention to transition, or who have completely transitioned. Where necessary, I will make the distinction.
I will also attempt to address the situation in the locker room and on the playing field, i.e. in both sport participation and gender segregated facilities. Let's see if I can do this without "man" or "woman". If I refer to "male" or "female", I mean the biological terms. Also, I will continue to discuss primarily the mingling of transwomen with ciswomen, but most of the arguments apply to mingling of transmen and cismen, but generally with less significance.
I will address your main argument now.
1. Women don't feel comfortable undressing in front of men.
Ok. So where do we go with that? I am a man and I don't feel comfortable being naked in front of men either. Most people probably dont feel comfortable being naked in front of strangers. You know the 'oi oi bums to the wall boys' brigade dont feel comfortable changing with gay men too.
The question is to what extent do feelings of discomfort allow us to dictate the rights of others.
Also note that you are now classifying transwomen as men apparently. Which is problematic.
In accordance with the terms of the debate, I will rephrase my original (which Archie has already rephrased).
1. Cis-women feel uncomfortable undressing in front of cis-men.
A lot has already been said about whether or not that ought to be the case, and whether or not it is instinctive or a cultural artifact. I won't elaborate here. The fact is that most of them do indeed feel that way and that isn't going to change.
As Archie points out, there are also varying degrees of discomfort experienced by people undressing in front of cispeople who share their own sex, and this is unquestionably true. If given an opportunity, most people would opt for private changing areas. However, there are economic considerations. Simply put, it is more expensive to pay for privacy. When we look at patterns in private gyms, we can see what people are willing to pay for. The vast majority of people will opt for segregated rooms, which they perceive as a benefit even if it comes at a slight cost, but will not pay for complete privacy.
The question of whether or not they ought to feel this way and whether we ought to accommodate their feelings in public (i.e. governmentally operated) facilities will be addressed in other sections below.
When it comes to ciswomen undressing in front of transwomen, the discomfort level is actually unchanged, because the transwoman is indistinguishable from a cis-man. The only differences are inside their heads, which cannot be discerned by the observer. (Note: This only applies to pre-operative or non-operative transwomen. This point does not apply to post-operative transwomen.)
Since this point is about comfort, not about performance, there is not much significance to the subject of athletic participation when it comes to the subject of discomfort, with one notable exception. In the sport of wrestling, close contact is part of the sport, and at least some incidental contact with intimate areas is inevitable. A high school cisboy is almost certain to feel extreme discomfort wrestling against a high school transboy. (If he doesn't, I would be a bit concerned.) A cisgirl may also feel discomfort wrestling a transgirl, apart and distinct from whatever advantage the transgirl might have as a result of her male biology.
2. Its discrimination against men, not trans folk
This fails the sniff test for me because it is not men who are impacted by the discrimation, but transwomen. You could make an argument that excluding gay men from locker rooms is not discriminating against gay men, but against people who have sex with men. I think (hope) you would agree this would just be clutching at wordplay straws.
The ciswomen are trying to exclude cismen from their spaces, and in order to do so, they must also exclude anyone who is indistinguishable from a cisman.
3. There's no reason to exclude black people
Indeed. But do you agree it was wrong to exclude them? Even before society had worked through it all? Because it wasnt fair to black people to exclude them without a good reason or to assume that exclusion is right until we can show it isnt. It wasnt right that black people were segregated ever because some white folk weren't comfortable sharing.
I would agree that it was wrong to exclude them. More importantly, this sort of "wrong" was not merely a moral wrongness, but a factual wrongness. We can look objectively at the difference between star bellied sneeches and plain bellied sneeches and conclude that regardless of whatever significance sneech society placed on the stars, there really wasn't any difference. Likewise with dark or pale skin tone, which is what Dr. Seuss was really talking about. There is no objective difference.
In the case of sex, the differences are significant. A male is capable of impregnating a female. That matters. There are instinctive sexual responses associated with male and female bodies. That matters. As we all, know, sexual orientation is not the same as gender identity, in which case a transwoman might be a lesbian. If that is the case, then the ciswoman is undressing in front of a stranger, who happens to be large enough to overpower her physically, penetrate her against her will and impregnate her. Even if the physical surroundings or social setting is such that such an attack is unlikely, there is an anxiety associated with proximity to a naked cisman due to that possibility. Most people would say that such anxiety is perfectly natural and understandable. Since the transwoman is indistinguishable from a cisman, the anxiety level would not be diminished in the presence of an unknown transwoman. The anxiety has, as its source, genuine differences between biological males and biological females.
In the case of a known transwoman, (i.e. one personally known to the ciswoman), there may be some sort of relationship that puts her mind at ease and would make the ciswoman more willing to undress. However, in that case, it's her choice. Oh, and there might also be a relationship with the transwoman that would make the ciswoman say that no way in Hell is she going to get naked near that specific transwoman.
I am stressing the possibility of physically overpower as significant there, and it is, but some people might make the mistake of then thinking that there should be no problem with a transboy undressing in the presence of cisboys. The transboy cannot overpower them and penetrate him. No, they can't, but there might be other issues, especially in high school. The possible consequences of putting a naked transboy into a room with one or more naked cisboys is left as an exercise for the reader.
I am stressing the most extreme differences and possible problems between trans and cis variants here, but in reality there are a whole host of issues, some emotional and some physical, all related to the fact that males and females are capable of sexual intercourse and reproduction. These differences are significant, unlike the difference in skin tone that used to be considered significant.
In the case of sporting participation, there, the difference are also obvious and significant. The transwoman has a clear advantage, not based on any societal actions, but purely on biology. Some of those advantages persist even after transitioning. I will leave it to specific governing bodies of specific sports to decide what rules ought to apply to transwomen participating in the "women's" category, but there would be no injustice done if all transwomen, regardless of transition state, were simply banned. There may be arguments for allowing them to participate, possibly with restrictions or handicaps, but it would not be unjust to ban them.
Maybe some would not perceive it as an injustice to allow them to participate unrestricted or unhandicapped. It might not be a "justice" issue as such, but it would simply make the competition insignificant. A transgirl (or transwoman) with marginal athletic ability will win a competition against a cisgirl (or ciswoman) with exceptional athletic ability.
Archie's original post had points four and five as well, but I don't think there's anything really significant that is different from the points above. I think my answers above address those points as well, but if there is something I missed, I'll address it later.