Not the erasure of the word 'woman.' The expansion of the word, to include those women who have been ignored, rejected and abused.
The word 'cisgender' is necessary. Quite often I hear a clueless person contrast trans women to ... you know, real women. Regular women. Normal women. Those imply that trans women are irregular, unnatural, abnormal. Which is hella insulting.
Trans folk have existed since humans have existed.
We could create a subcategory - a subclass. Women 'beta'. Wannabe-women. 'Almost' women. But then what? I'll tell you what. Soweto. Gaza. Jim Crow. Separate but equal. And abuse and discrimination. That's what happens when you declare a category of people "separate but equal."
The humane option is to expand the definition of "man" to include trans men (which btw IS happening. Your point about this not being a thing with the word 'man' is not a reality). The humane answer is to expand the word 'woman' to encompass BOTH cis women and trans women. The humane answer is also to expand the dyadic construct of humans to allow for non-binary folx. Because they exist, and we've ignored them for centuries in Colonized society. (They were respected in most pre-Colombian cultures.)
The idea is not to reduce the rights of cisgender women. The idea is to allow trans women to breathe. Integrating bathrooms by race did not erase white women. Integrating bathrooms by gender variance will not erase cis women.
When you say "people are confused and frightened," you're so close to actual introspection. Say instead that you're confused and frightened. And ask for clarity, rather than spreading your confusion and fear.
