Obligatory remark: Let's also put the cars on metal rails instead of on asphalt to reduce rolling friction. And maybe instead of small cars, we could use large wagons or chains of those that could transport a lot of people at the same time.
What you really want, to replace private cars, is SkyTran: tiny cars for 1-2 people, suspended from metal rails (which have a maglev effect to eliminate friction altogether), that are completely autonomous and take riders directly to their destination within a metro area. Each one operates independently, so you don't have to worry about what other users are doing or waste time going to stations that aren't your destination.
IMO, it's shameful that we haven't done more to explore PRT (personal rapid transit). It could really revolutionize city and suburban life and eliminate the need for so many people to own cars, while also saving people so much time with commuting (since such a system, fully built-out, would have far higher capacity than our current road system).