I hope at least that the people who say penises are shocking to women would not suggest that I should expose my daughter to multiple of them in the men's room?
Penises aren't in an of themselves shocking. I don't have a problem with nudity as such. I'd prefer to see more wedding tackle on TV and less gory violence, myself.
That's not the issue. It's not "seeing penises are scary". It's not a prudish tendency toward puritanism that is the concern.
The problem is that these are relatively isolated areas, not under the direct observance of authorities, in which people are naked and vulnerable to others. Not all people present the same level of risk to others.
Data and resources have been provided multiple times, and it's been repeated many times, but you just keep ignoring it:
1 in 3 females has been sexually assaulted
1 in 6 females have been the victims of attempted or completed rape
90% of the victims of sexual violence are female
98% of the perpetrators of sexual violence are male
Most males aren't bad people. Most males aren't inclined to assault females. But some are.
And there's no way for females to know which is which.
And if that particular male happens to be one that is inclined to assualt females...
there is almost nothing a female alone with him can do about it. The best course of action for a female, across the board, is to simply never be in vulnerable and relatively isolated situations with adult males.
When you take your daughter into the mens room, some of the men might feel uncomfortable, and your daughter might feel uncomfortable. But none of you are going to feel intimidated or at risk of sexual assault. Your daughter isn't a physical threat to the men in there, and the men in there are unlikely to see you as a risk of assault. If nothing else, there's a reasonable likelihood that other men could fight you off if you lost your mind and decide to try to rape one of them. There's at least a reasonable likelihood that they could give as good as they got.
If a woman takes her child (of either sex) into the men's room, some of the men might feel uncomfortable, and a female child might feel uncomfortable as well. But pretty much none of the men in the room are going to feel intimidated or at risk of sexual assault from a female. A female is, in nearly every case, not a physical threat to a male. In almost every single case, a male will have no trouble at all fighting off an aggressive female.
When you, a male, take your daughter into the womens room, most of the women are likely to feel uncomfortable, but your daughter will probably feel very comfortable. In addition, most of the women will acknowledge that you could be a real threat of sexual assault, and are likely to feel intimidated whether you think of yourself as a threat or not. Because... if you were to happen to be one of those males inclined toward sexual violence, there is not much the females in the room could do about it. You're extremely likely to be bigger and stronger.
Now... there's a reasonable likelihood that if you are accompanying a young girl, most of the women will assume that you're chaperoning her, and they'll assume that even if you are someone inclined to sexual violence, you're not likely to do so in front of your daughter. And based on that risk assessment, they might decide that your presence is tolerable and low risk.
Don't make the error of assuming that the subconscious risk assessments performed by women in that specific situation can be generalized to all situations. And don't make the error of assuming that the risk assessments done by females are flawed or exaggerated because you know that you specifically are not a risk.
There's no discussion to be had then. You don't allow for any other view to be presented, because disagreement is assumed to be evidence of bigotry.