1) Being driven around is the best transportation mode for most of the US. It's very comfortable, private, fast, and point-to-point. It stops working well at very high density, but that level of density is only seen in a few places in the US. I'd like more people to live in dense areas but for the foreseeable future self-driving vehicles are going to be the best solution for most trips in the US.
2) At very high densities it's true that cars can move fewer people per hour per 10-foot lane than other modes and so you run into congestion. But that's measured with the current vehicle fleet and human drivers. With high autonomous vehicle penetration you could implement congestion pricing that encourages high throughput vehicle design. That means private vehicles that are much much smaller (think Isetta-like design) that can follow at very short distances. Along with the elimination of on-street parking we could see a many-fold increase in road throughput.
3) At even higher density levels the same congestion pricing mechanism would encourage people to use microbuses that would operate similarly to Uber Pool. Compared to today's busses they would have equal or greater throughput, be point-to-point or nearly point-to-point, dynamically routed, cheaper to operate and faster.
4) At the very highest density levels it's true that nothing can match the throughput of the subway. As others have mentioned, AVs are a great way to connect people to the subway. Many trips intersect with the highest density urban core for only a fraction of the journey. More people would take the subway if they knew they could get to and from the stations easily and quickly. AVs let you mix-and-match transport modes more easily.
Cities should start engaging with vehicle manufacturers to start getting these high density vehicle designs worked on and figure out the congestion pricing mechanism to properly incentive their rollout.