Cities are not designed around the rich or the ultra-rich. Transportation systems are not designed around the rich or the ultra-rich. If all of us had infinite money we would just use private jets or something, but that doesn't mean that jets are strictly superior to cars... this is an apples to oranges comparison.
As someone who's lived in Chicago, Seoul, and South Carolina, I can say that our public transportation systems are wholly inferior to Korea and Japan's public transportation systems. It's not strictly a matter of money or not, it's a multifaceted cultural topic.
The whole idea of ranking transportation systems by what rich people do or don't do makes absolutely zero sense.