The point of gendered pronouns is that sentences very often have two nouns: a subject and object. If you have only a single pronoun, then any sentence can only use a pronoun to unambiguously refer to one of those. If you have two pronouns and you randomly and evenly distribute them across various subjects, then it's fairly likely that the subject and object will get difference ones and you can now unambiguously use pronouns for both.

Since a very large fraction of human speech pertains to other humans and since human sex is a roughly 50/50 distribution, it makes some sense that we used gender as our mostly arbitrary thing to assign a pair of pronouns too. We could have used, say handedness instead, but with only ~10% of people being left-handed, the odds of a sentence being about a righty and a lefty is much smaller.

Pronouns are basically like special variables in a REPL that let you refer to the most recent expression. It's nice if you can refer back to a couple of them.

Hence why forcing my team to learn Hungarian has really been a no brainer and will really start to pay dividends in the next decade when we finally reach a working professional proficiency

No, that's not the point. (Most languages do make subject/object fairly clear.)

The point of pronouns is to have a short alias to refer to a previously introduced noun.

Though now that I think about it age is also potent - you get the same rough division w.r.t relative age (younger vs older than the speaker), and you can divide the population roughly into quarters with ease and accuracy. Do miss and ma'am count as pronouns here? They code for age as well as gender

Hair color and height dont really have the same sort of dividing power broadly speaking.

Still, your line of thought there about what a pronoun should capture is intriguing.