Think you mixed those up, conservatives believe sex and gender are identical and don't differentiate. Progressives believe gender is a social construct, but depending on who you talk to that means the path forward is twofold and not everyone really agrees on what form gender should take in the future.
Both. Just like race. Its subject to change based on changing concepts, but are regardless of which version of the social construct is used, race and gender are generally based vaguely on immutable things.
Gender is a social construct designed around sexual dimorphism
Females of the species are equipped to and (before the development of birth control) likely to carry children, which inconveniences them physically for several months. Historically this was somewhat frequent. This caused a selective evolutionary pressure to concentrate traits compatible with said inconveniences on the female chromosome, and concentrate traits less compatible on the male chromosome. This created the kernel of gender roles, which themselves evolved over the years under the resulting social pressures of gender interplay.
Consequentially, women traditionally fill roles that can be accomplished while pregnant and/or breastfeeding (cooking, cleaning, childcare, weaving, sewing, etc) and men traditionally fill roles which are particularly difficult to do while pregnant and/or breastfeeding (hunting, farming, other strenuous labor, etc). These were reasonable adaptations that were broadly useful for quite some time. So in a sense, we're born with it.
Recently, developments in housekeeping (breast pumps, formula, automatic appliances, public schooling and childcare, affordable industrial textiles, etc) and labor (the transition from physical to mental work) have made the biological differences between males and females less relevant in fulfilling social roles. What's more, social roles have changed so much anyway.
Personally I think we're getting to the end of the usefulness of gender as a social concept.
If gender is fake why is gender identity so important? If gender is fake then how some people feel they’re the wrong gender? If gender is fake why is it so important to learn about someone’s gender because using the wrong pronouns will give them trauma?
Get your story straight on gender challenge for leftists (IMPOSSIBLE)
I always want to learn more about things. To be totally honest with you, what I'm having the most trouble understanding in the current gender discourse is how can gender be both a social construct/abstraction (in the famous words of Judith Butler "an imitation without an original") but then also gender identity is a deeply-seated innate feeling that people have that then enables the feeling of "my gender is wrong and I need to change it."
I really don't want to be transphobic or even enbyphobic or anything and I will use whatever pronouns people want to be nice. But just on an epistemological level, I'm having trouble understanding it.