"Tall" is an acceptable descriptor for a person, obvious to all and not usually very sensitive. "Trans" isn't like this and remarking on "how trans" someone is carries a very different meaning than remarking on "how tall" someone is.
Maybe it's still acceptable or useful to accept "trans women" as their own group, but eventually that will probably go away and "trans woman" will become a dated and bigoted term.
You may find yourself shocked that you ever used the term "trans" in the same way that it's now shocking that black people were once referred to as "colored".
Or alternatively, being trans might become an identity badge of honor, in the same way "gay" has in many places.
My point is that these things evolve over time, so either people are going to be allowed to make mistakes and have debates and figure things out in public, or we have to only use useless tautologies to describe people.
When you tell the GP (who claims to be pro-LGBT) that they're not pro-LGBT because they're struggling to speak correctly on the topic and they used the "wrong" language, you might not be having the effect you intended.
In this case, the GP mistakenly believes the dated idea that "gender" is a biological term, but enlightened folks are aware that gender is now a social construct. I think they simply meant "biologically male people are not female" instead of "trans women aren't women". The language is confusing because it's evolving, but getting everyone on the same page is going to require a lot of public errors that would be better met with debate than shaming.