Hello ciderguy

A transgender woman is a woman who is trans. like a big woman is a woman that is big. So if a woman is trans that means she was assigned male at birth but isn't male, therefore trans.

To the question if we just could say man and woman: Yes and no.
The thing is in a conversation that has nothing to do with them beeing trans it would be great to not mention it and to aknowlege that they are no less male or female because of that.
But there are topics where it's important (example: bathroom bills) because it emphazises that we have very different experiences in how people treat us. So in this context it matters.

I hope I could help you