Yes, gender is synonymous with sex, and, yes, there are but two genders. What a confused fellow like you means by "socially-constructed gender identities," are what are called, "gender norms." That there are gender norms is completely uncontroversial and demonstrable. What is completely unsupported is the assertion that someone can be "born into the wrong body." The irony of this, of course, is that for decades feminists worked overtime trying to overturn the idea of "female brains" and "male brains," in addition to the fact that the signals for whether someone, usually a child, is "trans," is determined in large part by how much that child or person conforms to particular gender norms. These sorts of obvious contradictions, of course, escape the dimwitted gender theorists who manufacture this claptrap.
Who are these "experts" you mention? Biologists? Please share with us the biologists willing to go on record and state that there are more than two genders. Whoever they are, they certainly won't be furnishing science to back up this claim.
Yes, one explanation is that all these folks were "born that way," and only now feel the freedom to reveal their "true identities." This could be the reason that countries like England have seen something on the order of 1000% increases in the numbers of people, particularly children, coming out as trans. Unfortunately, that explanation doesn't seem to hold up well. First of all, if it were biological, we would expect an at least somewhat balanced distribution of males and females declaring this. Instead, what we see is larger and larger disparity in the number of women declaring that they're transgendered, versus the number of men declaring themselves to be transgendered. Given that there are even stronger norms from the past discouraging men to identify as women, we would logically expect the reverse of what is happening. Second, with all these women and, more specifically, young women and girls, declaring that they were born in the wrong bodies, we see clusters of these declarations in friend groups. It's quite clear that what is happening is driven by environment, not biology. Teenagers and younger girls have friends who declare this, and soon the friends of the "trans" person discover that they too are trans.
Encouraging young women and girls to declare this and helping to push them towards unscrupulous and politically-motivated health professionals is not compassionate. Quite quickly, these girls are pushed to take puberty blockers and even hormones. The hormones have some irreversible effects. We're only beginning to learn of the effects of puberty blockers on them, but sterilization appears to be a consequence for some. Mind you, a large number of these poor kids and young women will regret this course of treatment as they get older. We have no longitudinal research on the effects of pumping kids with these things. As others have said, these kids and young adults are virtual guinea pigs. You can cloak your mouthing of these mantras as compassionate, but advocating and supporting these treatments is clearly not compassionate to reasonable people with moral compasses.