Re: TL;DR: A few details that most of my fellow commentards seem to be missing...(@ AC)
You forgot that as cities get dense, land prices soar up, pushing all but -sometimes- the very wealthy out of city centres. So most of the people that want to [work, buy, study, go to a doctor, ...] there have to come from outside those city centres, worsening the traffic problem and negating your point.
There are many other factors involved and discussing them all would probably take weeks, but in my opinion it's clearly a chaotic problem. Many complex balances interacting with each other. We know some of the attractors, though, and one of them is that a good public transport system shows a stubborn tendency towards making things better both for citizens and government.