> A fair percentage of people do not accept the concept of transgenderism, and instead see a transgender woman as a mentally ill man
A not trivial amount of the objection is because of the singular 'they' and 'their'. I'll cop to being confused by sentences leveraging those formations. One can appeal to etymology, but even knowing the etymology didn't prevent my confusion.
"Be kind" is a good rule. And now there are also practical concerns to be sensitive to.
Point being, it's more complicated than accepting new gender models. And whatever peaceful solution is possible, I don't see any that don't involve decades of patience as folks get comfortable with new grammatical constructs: gender agnostic singular pronouns other than 'it', sentences containing 'their' not implying plurality, or something along those lines.
Keep in mind that the U.S. still uses imperial measurements.