I don't see this as a left/right dichotomy at all. there are as many authoritarian left wing regimes as right wing ones, so a simplistic 'authoritarian=right/libertarian=left' doesn't hold water, and feels like a dig at those of us with both left wing and biological realist views. Boris Johnson is a libertarian, but hardly left wing, and Mao was authoritarian but not on the right. But we've had that discussion before.
I agree that there have been occasional (recorded) instances of people 'passing' as the opposite sex, but that is not what Smileless is asking. What is now called 'transgender' used to be known as 'transexual', and that shift, I think, heralded the start of the misogynist approach to male entitlement. Sex is easy to prove in the vast majority of cases, but so-called 'gender' exists in the mind, so cannot be proven. It is, anyway, a nebulous concept, which changes across culture, time and place.
There was a time when it might have been possible to have men and women, girls and boys, all able to choose the behaviours and preferences that suited their personalities with nobody telling them that a preference for pink is 'female' and liking Meccano is 'male', so if they enjoy these things they are 'in the wrong body' and should 'transition'. When my children were babies and children (in the 90s) I don't remember any of the angst that children now are put through. Everyone I knew, plus the staff at the schools they went to, were tolerant of boys who wanted to be princesses and girls who wanted to be Power Rangers. They were encouraged to fulfil their own potential regardless of sex, and 'gender' seemed to be becoming less relevant. I don't know why that didn't continue to be encouraged, but at some point the 'authentic self' thing became fashionable, gender stereotyping really took off, and brought with it the mess we are now in.