Sure, it is a "free country", but unless someone has a particular insight into how someone else thinks, going with the original birth gender is nothing short of judgmental.

What the heck has "how someone else thinks" got to do with it?
A discussion about self-perception and how others perceive that perception would seem heavily important here.

I know, right? And yet the usual attitude from trans rights activists is that any such discussion is completely unnecessary and everybody should just take for granted that self-perception is the defining criterion. So let's have that discussion.
If a transgender woman didn't feel like they were the opposite gender, this wouldn't be a discussion.
What does "feel like they were the opposite gender" mean? Does it mean "feel like a person of the opposite gender feels"? If so, they have no way of knowing they feel like that -- a transwoman has no experience with what being a biological woman feels like. No biological male knows that. So when one says "I feel like a woman", it's pure speculation. A transwoman could more accurately say "I feel differently from cismen; I interpret my feeling as femaleness, but for all I know femaleness might feel very different from what I feel."
If some (most?) women didn't have a propensity to regard a transgender woman in a private space in such a way, this wouldn't be a discussion. How we think and feel is a keystone to this conversation.
Sure, but that's how
we think, not how
someone else thinks. Most women have a propensity to regard a transwoman in a private space as the opposite gender, not because they have any particular insight into how the unexpected entrant thinks, but from observing secondary sexual characteristics and inferring primary ones.
Some want to ignore the psychology and just hold directly to visible biology (certainly ignoring neurology), but that would seem daft.
Let's unpack that. "Want" isn't the issue here. Some (a word that here means the vast majority)
do ignore the psychology and just hold directly to visible biology (certainly ignoring neurology). So what exactly are you calling "daft"? That they do? That some of us take note of the fact that they do, and we don't decide whether they do based on wanting, or based on evaluation of daftness?
The usual attitude from TRAs seems to be that it would be a better world if women categorized based on the person's psychology, and therefore we should all make believe that women already categorize based on the person's psychology, and eventually women will be retrained enough that the pious fraud will become reality and the better world will be achieved. I'd need a crystal ball to say if this strategy will work, but I can say right now that this strategy is intellectually dishonest. The incessant equivocation between "gender" and "gender identity" is well-suited to sweeping the dishonesty under the rug so the TRA's can still feel good about themselves while they do it.
I don't need to know how Tenzin Gyatso thinks to judge that he's not anyone's reincarnation, and if you find that judgmental of me, that's nothing short of judgmental of you.
Reincarnation is a spiritual concept invented by people. Gender is a taxonomical like concept based on a broad baseline of outwardly obvious and inwardly (to organ level) discernable attributes.
Right. Both are categorizations invented by people that involve factual claims -- claims on which how the person thinks logically has no bearing, even though people's feelings about how they themselves are categorized are important to their self-image. It might hurt a Buddhist's feelings to have others disbelieve the claim about being reincarnated but that's no reason to believe. It might hurt a transman's feelings to have others disbelieve the claim about being male but that's no reason to believe.
We know now that gender isn't as simple as that, because neurology.
Who you calling "We", white man? No, I don't know gender isn't as simple as that, because neurology. How does neurology imply gender isn't as simple as that? You appear to be assuming your conclusion as a premise.
"Phase 1: Steal underpants. Phase 3: Profit!". What's Phase 2?
What I don't take seriously is the religious belief of gender ideologues that an individual's inner feelings trump reality. Gender is a social construct. What the criteria for the genders are are up to society collectively.
That is a peculiar statement for you to use. It admits that "gender" isn't steadfast and carved in stone.
"Peculiar"? What's peculiar about it? "Admits"? What's an admission about it? How do you figure this is any sort of concession? Where have I indicated "gender" is steadfast and carved in stone? Were you taking me for one of those "man and woman just mean biological sex" hardasses? That's Emily and seanie's thing, not mine.
If you're arguing that gender being a social construct that isn't steadfast and carved in stone somehow supports the contentions that self-perception and neurology and "how someone else thinks" determine people's gender, show your work.
There could perfectly well be a society that doesn't give a toss about biological sex*, and classifies people by counting their teeth. It could have a language with three pronouns, "he" meaning "the above 32-tooth person", "she" meaning "the above unusual tooth-count person", and "it" meaning "the above toothless person or nonhuman". It could even unsteadfastly evolve its categories and language to change from "current number" to "number at age 21", and reclassify old people who've lost teeth from "she" to "he". But my mom never grew wisdom teeth in the first place; if she lived in that society, no amount of identifying as a 32-toothed "he" would ever make her one. "Social construct that isn't steadfast and carved in stone" in no way implies "self-perception matters". That's a non-sequitur.
(* In principle. In practice, giving a toss about biological sex is a human universal, found in every society anthropologists have checked.)
That it is possible to adapt with better understanding on what provides a gender beyond organs.
Certainly. Emily and seanie's contention that "woman" means "adult biologically female human", like any other word definition, is a
theory. It might be wrong, and we might replace it with a better theory. It also might be right currently, but become wrong in the future as our language and culture continue to evolve. The point, though, is that like any other theory, their theory is
falsifiable. If you think it's wrong, present evidence that it's wrong and propose a different theory that better retrodicts empirical observation -- or else think up an experiment the competing theories predict different outcomes from.
Understanding transgenders, their mindset, how/why they think, their experience will help provide a better manner in which to move forward with inclusion and how best to achieve it.
Certainly. But it's unlikely that discovering a better manner to move forward will produce evidence that transgenders actually are the gender they think they are.