(this is a genuine curiosity, I'm not trying to be ignorant or start drama)
Most Helpful Guy
I was extremely skeptical about this, but I think the answer is that there are two types of "transgender" people.
One is the type that just doesn't like being associated with the behavior patterns that is socially defined and forced upon a particular sex. For example, you dislike how you are prevented from wearing the fancy colorful clothing and jewelry and whatnot that is created exclusively for females. Considering this type of socialization would be missing in your hypothetical scenario, and assuming every sex could do whatever they want as they are segregated, they would be socialized not to look for wanting to be the other sex, because the differences they desire and the restrictions they loathe would be gone.
The other however is those who are legitimately feeling that there is something "wrong" with how they are. I started understanding this after watching this video
Yes I know it's 15 minutes long, but hey, you asked. Take the case of David Reimer; his penis was removed because of a botched circumcision (yeah I know what you're thiinking, shh), and was an attempt to socialize him to be female. It did not work; he did take on the socialization elements that are associated with males. Nobody told him! He did that on his own. Essentially, the brain IS wired to be a particular sex, and takes the elements for that type of behavior from its surroundings. Which means, if there is such a wiring - a "mapping" for what one is meant to be like and if it's all fine and dandy - then that wiring can also be misaligned and broken. Similarly to how there are people who are born blind or are born with underdeveloped genitalia - hormones are imperfect, nature is imperfect.1