It's about the women who come to shelters who are afraid of men, for good reason. Men lie. Men who are stalkers lie. It would not be a bit difficult for a man to come to a shelter and SAY 'but "i'm a woman!" and gain entrance for the purpose of coming after someone.
This paragraph is ridiculous on multiple levels.
1 - Dangerous stereotyping of men as inherently a threat to women in all situations.
2 - Confusing trans female with male (again, for some reason).
3 - Women are just as capable of doing the exact same as everything you just outlined.
4 - Shelters have checks and balances in place to ensure the people who use them are safe.
5 - This situation could just as easily happen whether or not a shelter allows trans people to use it (for example, a man could attend the shelter saying that they are actually trans
male, but can't use a men's shelter because of their biological gender).
Is this a little 'out of the realm of possibility?" Not really. Women who must resort to such shelters are dealing with men who will do just about anything.
If that's the case, then what difference would it make whether shelters allow trans women into them or not?
So...a man who cannot prove that he is properly diagnosed and on a treatment plan to change his physical gender would be, were I in charge, told no. when he can come with the paperwork, ok, but based only on his word?
No.
So you'd deny access to somebody seeking assistance who didn't have the foresight to grab a copy of their medical records because they're fleeing an abusive relationship because you think trans women are dangerous despite a complete lack of evidence, and you'd potentially leave that person in greater jeopardy by forcing them to use a men's shelter (where they are substantially more likely to be subject to abuse) or returning to their abusive relationship.
On top of this, you would also have the apply the exact same rigmarole to every cis woman who attended the shelter, because you would need to investigate whether or not they are actually trans women before you gave them access, so they are also subject to all of the above issues at a time when they desperately need access to sanctuary and security.
And this is why you don't run a shelter, I presume.