Not quite. The thing is, gender identities is a complex topic with vastly different (political) opinions. To objectively explain it, you would have to present the different views and how some people believe this and others believe that. However, children don't do well with these multiple realities, most would most likely ask you "Well, which one is true?". That, however, is a political opinion.
I therefore think it should be up to the parents to educate their kids in that regard, according to how they see fit, and not up to the institutions, at least not until kids are old enough to reflect about their political views and form their own opinion upon being presented with the topic in an objective way.
Except, one viewpoint is objectively not reality. That's the whole point of this post, one of the two main political opinions hold the beliefs that:
ALL trans women and gay men at pedophiles
ALL trans men and gay women are just confused and haven't found the right husband yet
Being trans and transitioning turns you into a sad, pathetic mockery of the other gender who is destined to kill themselves from the shame of it all
ALL Queer-adjacent children (under 18) are also confused innocents who have been lead astray by evil forces trying to corrupt the new generation
None of which are true, in case it wasn't clear. So no, it should absolutely not be left up to the parents to teach, because when that happens, a solid 30% of (American) childeren are "educated" about a complete un-reality that only exists in the minds of hate-filled bigots, and terrifies all us queers back into the closet.
Those "beliefs" are in no way representative of "the other" political opinion. For example, here's my opinion on the matter (and I disagree with every one of the 4 conservative extremist "beliefs" you posted about) so you can understand what political differences I'm referring to:
You can love/marry/have sex with anyone you want to, it doesn't matter what gender. (Most people would be onboard)
There are (maybe appart from people who were born with a different number of chromosomes) 2 genders (already some political disagreement here)
Feeling that a gender doesn't belong to one doesn't change the gender. Just like feeling like I have blond hair doesn't change the fact that I have brown hair. (This is were people heavily disagree and you can't easily present either opinion on this as the sole truth).
It might be important to note that this is not meant as an insult to you or to anyone and it doesnt mean I dont respect trans people.
I'd be interested to hear your thoughts on this...