gender identity is highly personal
Yes. So personal that it makes no sense to call it GENDER (a category which is a collective group) IDENTITY. It is INDIVIDUAL IDENTITY.
So why should a person's INDIVIDUAL identity cause them to erroneously believe they should be included in a GROUP? If you don't possess the characteristics of that group then you are not part of this.
I have XX chromosomes and ovaries. (for example) Anyone who also has XX chromosomes and ovaries objectively shares something in common with me. We are a group. There's a name for us.
A person who objectively doesn't meet that description has no earthly reason to use the word that is shorthand for people with XX chromosomes and ovaries.
If there is some other characteristic they possess, that matters greatly to them, let them state what that characteristic is in plain terms and other people will either also possess that characteristic and be part of that collective group, or it will be an individual characteristic unique to the individual.
If any woman genuinely believes herself to have some characteristic important in common with transwomen that distinguishes them both from men, then wonderful. Name it, form your group, fight for your rights.
Don't just take a word with meaning, empty it of its meaning and wear it like a bag you stole, emptied of its owner's contents and claimed as your own.
I don't know what, exactly, certain people believe that some women and some transwomen have in common that makes them profoundly different from men and in need of shared space but I'm happy to support the creation of third spaces for those people.
I personally have objective, factual common traits with all biological women, and I want to retain the spaces where I can be separated from men.