> My native language has no gender. Every time I see these conversations I wonder “why not do away with the gender?”

Yes, many languages have no gender or have a generic option besides genders.

I always find this very annoying in Roman languages, where every object has an arbitrary gender, and to make matters worse it's not always the same. A word that is female in Spanish could be male in Italian or Portuguese. It trips me up a lot and it's annoying because the gender is just some arbitrary thing there, it has no meaning because objects have no gender. If we're talking about people I can understand but things?

But languages are not very logical. They are something that melted together over centuries out of habit.

I wonder how you would feel about, for example, Slavic languages. There, you leak information about the gender even about speaking about yourself in first person (at least in the ones I have some idea about). Example in Polish:

"Spotkałem kolegę" - "I [male] met a [male] friend."

"Spotkałam kolegę" - "I [female] met a [male] friend."

"Spotkałem koleżankę" - "I [male] met a [female] friend."

"Spotkałam koleżankę" - "I [female] met a [female] friend."

Notice the different ending (em/am) depending on whether a male or female is speaking.

> The patriarchy - along with religion - wanted this gender line and specification in every context

Did they? English used to be gendered but lost it ~1000 years ago.

Languages evolve in odd ways and don’t think the patriarchy or religion have much to do with it.