"Where does HN get it's perfectly compatible partners from?"

Truth is women in our culture are still somewhat of a status symbol. My wife is essentially perfectly compatible with me that was achieved by not simply selecting the most physically attractive person who would marry me but instead finding someone who actually is a decent human being. Her ring cost under 50 dollars. She wants me to spend the short time we all have on earth with her, not working to buy her shiny rocks, a overvalued house or an expensive car so she can impress her friends. I wouldn't have it any other way.

If people want to spend their money on a ring then I have no issue with that. Personally my wife is quite frugal about signs of material wealth (raised in a family with money that drove old Honda's etc) and had very strong opinions about not wanting an expensive ring. Instead we spent our money on a honeymoon exploring Asia and Europe - I wouldn't trade that time for any ring. Travel may be a privilege but teaches you a lot more about the world and the person you want to spend the rest of your life with.

So why does only one partner get a ring? Why, typically, the female one? In same-sex marriages, how do partners determine who gets the ring? Flip a coin? Damn.

The answer to all your questions is: it depends on the partners and what they agree to. But on the most part, it is driven by cultural momentum - the same reason men's formal clothing includes a tie. One could equally argue "One does not need to wear a tie to go to work", but there are people who love ties and how they look wearing them.