Americans own cars because most of them live in single-family houses on large plots of land, and that doesn't make public transit for daily commuting a realistic possibility. In Paris car ownership is very low, maybe 1/3 of adults, but in rural France the car ownership rate is easily 95%+. I haven't seen a single developed area in the world that has violated the rule that low density = high car ownership and vice versa.
The other rule that I have never seen violated is that the large majority of middle and upper income people do not want to live near low income people, due to crime or other reasons. In Europe, poor people live in the suburbs, so the middle income live in the city with high density housing. In the US and some other places (south asia), low income people live near the business center, so the middle income live in low density housing in the suburbs. These are for historical reasons and cannot be easily changed.
That is because social housing is everywhere.
Also fast trains means no need to live in London to work there.