A royal commission did a study on "Traffic in Towns" in 1963 and concluded it would be impossible to serve a city with London-like size and density with cars even if you put the buildings up on stilts and built multiple levels of roads an highways -- the problem is that you can't support a high enough level of traffic on highway offramps.

New York City is pretty close to the maximum limit of car capacity it can support. For one thing there are only so many cars that can travel on so many lanes through bridges and tunnels. If you tried to add more lanes, you'd have the same kind of problems connecting those to surface streets that you get in the "Jetsons in London" scenario.

Robert Moses did have some plans for urban removal on the southern side of midtown to accomodate more traffic (and destroy traffic demand) but it didn't happen.

(Note the response to the car in most American cities was urban removal -- you can often find a big empty space in a second or third tier city that has a 'civic center' plopped down on it.)

Manhattan looks pretty much the same today with all the cabs that it would look like with self-driving cars.