What you missing is firstly, no man is a type of woman. But even if you were to accommodate that, they don't want rights on the basis of that. They want rights on the basis of actual women.
Many women are more than happy to acknowledge a third space for this 'different type of woman', let's call them a transwoman. But that's not what is being asked.
When they talk about a type of woman, they mean like a blonde woman, or a tall woman. But there is no 'type' of woman that is a man.
So it's a useless description. Even if it was accepted.
And the thing you have to remember is it is women who are being exploited here. Men identifying as women want to occupy women's spaces, but only on the condition that there are women in there to access.
If all the women get up and walk away to a different space, the original space is no longer acceptable. The women's presence is crucial.
Because this isn't about spaces, otherwise third spaces would be acceptable. This is about validation.
And it is the presence of women and girls whether undressing, going to the toilet, or talking about their rape, that is crucial for that validation.
And, the more inappropriate the presence of a man is, the more intense is the validation.
Hence places like rape refuges, breastfeeding groups, stillbirth support organisations being targeted. Where the presence of men is obviously completely inappropriate.
If you can make women accept you there, the validation is 100%, and every other place just follows suit.
Ask a transactivist if they will accept a third space. Based on the fact they are a type of woman, it can be a 'type of woman' space.
The answer is always no.