"Lots of people accept transwomen are women."
They are about as correct as those who accept cats as dogs - two words with entirely different definitions.
Woman: Adult human female
Transwoman: Adult human male who "feels like" a woman.
Female: Of the sex that can bear young or produce eggs.
Male: Of the sex that can produce sperm.
"I think it's quite helpful to keep this in mind when thinking about motives and why people get so angry about the issue."
Sorry, that is not 'quite helpful' because I have no idea what you wanted to say there but I hope you are not getting angry or saying someone else is about to.
Personally, I think definitions are 'quite helpful' and that much of this debate would be unnecessary if we all took a moment to familiarise ourselves with the dictionary definitions of the words man, woman, male, and female.