This doesn't make sense. In the 90% of interactions where biology is irrelevant, gender is also irrelevant. The pro-trans side has so far been unable to articulate a definition of gender that makes it relevant in any interaction - especially any interaction where biology isn't relevant either.
That is where things like pronouns come in. Or treating someone like a lady (or like a guy). What that means of course, varies.
JoeMorgue makes the excellent point that he doesn't treat men and women differently. If his male friend suddenly declared to be a trans-woman, he would treat them the same. And it would be treating her like a woman. For him, there is no behavior modification required.
But most people make assumptions:
You walk into a party and see a group of women and a group of men conversing. You can make a guess which conversation topics are going to be most entertaining to you. You might be wrong, but you might make some assumptions. If you have an extra ticket to a Rush concert, and two acquaintances you'd like to hang out with, you are likely to assume the man will be more interested. (Though you might ask the woman first out of hope.) Lifetime targets movies based on what they think women value. (Yes, there's sex, but it's mostly romance.)
And yes, those examples are pretty superficial. What I really think people mean about being treated as their gender socially has a lot to do with just accepting them and not labeling them as freaks because you perceive a male body in woman's clothes.
I think the problem is that gender is hard to define. We can give examples of gendered behaviors and gendered appearance but not gender itself. I think gender is an aspect of personality, which is itself difficult to directly quantify. (We can list behaviors and assign them to personality traits similar to gender.)Without a coherent definition of gender and its relevance, the pro-trans side is very much not "mostly correct". It's simply Not Even Wrong.
My brother (PhD Zoology) occasionally ribs his wife (M.S. Psychology or Social Work) about his being a hard science and hers a soft one. I'm a biologist, but I took enough Psychology to recognize important differences. Much less in psychology can be directly measured. You can't directly measure personality, intelligence, learning ability, or even mental disorders. I think you will find many different and conflicting definitions of intelligence. It's not firm. It's debatable. Asking for a firm definition of gender as a psychological thing is a difficult ask.
I'm not really sure it's as important as you think it is though, because you can't perceive or react to someone's gender. You can only perceive and react to someones actions, behavior, and presentation, which may or may not be reflective of their gender or some other aspect of their personality. I can't react to your self-image. I can only react to the image you show the outside world. you can only that someone in a way that matches with their presentation. (Which is something the pro-trans side needs to think about.)
That said, I think gender roles and eventually gendered presentation will evolve away. Clothing, hair, and makeup are becoming less gendered. (Or at least the means are moving closer together even if the extremes are not.) Careers and hobbies are overlapping more. And it's becoming more normal to have close non-romantic friendships between sexes. (Possibly because of less segregation of interests.) So I kind of see a gender-less future.
But we are not quite there yet. (Except for Joe.