"The bottom line is that the normal majority are pushing hard to claim the word woman without considering the challenges of a trans person."

Nope, you have this almost perfectly backwards. Trans women are pushing hard to claim the word woman without considering the challenges or simply the realities of being a woman.

Women came first.

I have no doubt that trans women have to struggle to "sort themselves out." And I fully acknowledge that I don't know what that's like. But that in no way gives them the right to appropriate somebody else's identity. If a white person decides they don't feel white, well, that's nothing to do with me. But if they demand to be described, legally and socially, as black, I think black people will take issue with that.

A trans woman who is trying to "sort themselves out" starts that process being described as a boy or a man. Because that's what we've called male human beings since the dawn of language. During that process, they may decide they prefer to be referred to as a trans woman. Okay, great, trans women are trans women. This is factually and uncontroversially true. But the word woman is already taken. And it's taken by a group of people who face their own unique challenges.