From https://www.coe.int/en/web/gender-matters/sex-and-gender

Sex refers to “the different biological and physiological characteristics of males and females, such as reproductive organs, chromosomes, hormones, etc.” Gender refers to "the socially constructed characteristics of women and men – such as norms, roles and relationships of and between groups of women and men. It varies from society to society and can be changed.

So people want to be seen as the same gender as they feel they are, this seems reasonable to me.

Also from that same site:

Traditional gender norms are hierarchical: they presuppose an unequal power structure related to gender that disadvantages mostly women.

People don't think there should be an inherent advantage to being male, which I also think is quite reasonable.

A bit of nuance to what you say there...

LGBT people say gender (not sex, gender) is a social construct because the evidence points to this. How gender has been expressed has varied wildly over recorded human history (from customs to clothes to behaviours to jobs to everything else). In any given point of history someone's sex has been linked strongly to a particular gender expression, but the fact that those expressions vary deeply from culture to culture show they're socially constructed rather than purely biologically determined.

When you say they think they're not "important", I think LGBT do think gender expression is important. What's not important is squeezing into the two expressions that society traditionally had. Or welding yourself to society's expectations based on what genatalia you have.

History (for the most part) had two distinct gender expressions corresponding to the two sexes. But this itself was heavily influenced by society being tightly coupled to the biological reality and differences between men and women. Women had babies. Men were stronger. The gender expressions followed from that and you had to stay in the one society expected because that's what kept society functioning. Religion is a social construct that enforces this.

But as society has evolved we're no longer bound to these distinctions in the same way and the gender fluidity of people - which has always been there - is now able to express itself in more variety.

There are people born male who are far more comfortable living in society's 'female' behaviours and traits. And vice versa. There are men who are attracted to men and women to women. There are people born female who have deep seated psychological need for their body to be male. All these people have always existed it's just in the past they got sidelined as 'sinners' or divergent because society basically consisted of childbearing and hard manual labour.

Something being a social construct has nothing to do with its importance. There are many social constructs that are extremely important, and many that are less so.

Check out Judith Butler’s Gender Trouble. One of the best books I’ve read on the subject

First LGBTQ people are still people they don't have a monolithic view of the world.

Trans people usually do think gender is an important thing, that's why they transition. And for people who do think gender is a social construct (both lgbtq and not) why would they care if someone wants to change it, it doesn't matter anyway.

The point is in acceptance. You want to be perceived in a different role? Sure there you go.

It doesn't matter, it's all bs. But for many people it's hard to exist without a script to play. If someone feels more comfortable playing a different role who am I to say no

It's not difficult to hold two ideas simultaneously:

X is important (because people decided to make it that way)

X shouldn't be important (and wouldn't be, if people hadn't made it that way)

This might fit better in no stupid questions.

Gender roles no, it's not important to assign certain behavioral expectations based on biological sex.

Gender perception? That's a different question. Are you a man? What do you do to let people know you are a man, and do you care if they mistake you for a woman?

Okay, so, first, none of us are biologically masculine or feminine, because those aren't biological concepts, they're social ones.

People who are biologically male tend to be assigned masculinity, and same with females and femininity. And just because something is socially constructed doesn't mean it is not important — as a million memes will tell you, we live in a society.

I think the heart of the issue is who assigned the gender roles: The gender role you want to assign me isn't important; the gender role I assign myself IS.

Household responsibilities are important, but gender roles are of course made up.

All of this shit, including the words we're typing and the devices we type them on, is made up.

This, like absolutely everything, is different depending on who you are. Nothing, everything and anything is important to someone and not important at all to someone else.

This is a fundamental misunderstanding of what gender is and where it comes from.

Gender is an entirely social construct. You are confusing Gender and Sex.

Sex is typically XY or XX chromosomal combinations in humans, but there are far more than just those two combinations in reality. So there are more than two genetically defined sexes.

Gender has nothing to do with genetics and is an entirely cultural social construct. In most cultures it refers to what we call masculine and feminine physical and personality traits. But again, it's not a cut and dry thing. These traits vary by culture and so do the number of genders. For example, in Samoan culture they have a third gender called Fa'afafine. This is a long established part of their culture and fa'afafine are generally accepted and celebrated.