Your private taxi cab is never going to compete with the people-moving capacity of a bus, or a train. A road with 100s of cars on it, is a demonstration of how many people are heading in a common direction. If you want them to move faster in that direction, find a denser form of transportation that doesn't take up the space of 100s of cars.
Is parking an unsolved problem? I think in most cities it's a solved problem, and the consequence is traffic. But I would be curious of your take given that it's the question you asked. What is there to solve for parking?
edit: Also, Public Transportation, arguably, fixes both traffic and parking.
But ya, if we could eliminate parking, we have some more space for some nice things, but having lived it, it doesn’t solve the main problem at all if private transportation is still viable.
I'm having trouble parsing that sentence, could you clarify?
"[A developed city isn't one where poor people take cars but where rich people take public transit.]"
The problem is that people need a place to store their single-occupancy-vehicles. which means that each individual with a car takes up a permanent ~100 Sqft. They also need places to put that car. If you remove the parking lots, and the roads, admittedly, you'll create a more walkable city that means people don't need cars. But people will still want something for longer distances. And that something is public transit.