I think history has romanticized car-less cities. Horses spook easily, and kill people in the process. Massive quantities of animal feces and dead animal remains would overwhelm the minimal city waste-cleanup services. The constant sound of horseshoes on non-dirt roads is worse than normal car noise. Cars solved real problems of city life.
I think a better argument is that 'everyone' has a car, but in old cities, far from 'everyone' had a horse - they were for businesses or the wealthy only.
I would love a car-less city where the only vehicles are public transport or commercial delivery vehicles, with cycling being the only allowed personal transport.
I think this vision is shared by some of the town councils in big european towns.
For example in Paris, the current mayor (who has been in office since 2001) has made it his mission to make the city as unfriendly as possible for cars to discourage people to come to Paris in their cars.
This includes massively removing parking spots, making parking expensive all over the city, removing lanes wherever there are roadworks (to add sidewalks, bike or bus lanes, trees...) and banning cars from some of the most important streets in the city (most notably the embankments).
So far it has stopped the growth in car usage in Paris. I don't think people are ready for an outright ban on cars yet, but this surely seems like an effective strategy to prepare them for it. Maybe when the Supermetro is done and suburbans can more effectively use public transportation the city will move in this direction.
However : do you consider taxis as public transport ? What about Autolibs (shared electric cars you can pick up all around the city) ?
Edit with some stats :
Parisians overwhelmingly use public transport (63%) and not their cars (13%) to go to work. More people walk than take their cars (14%). So that would be an indication of success, however a brief walk outside at 6PM will help you see that suburbans still massively come to Paris in their cars :)
I would like to say no, but I do see the need for personal transport on occasion. I would need to think this through more, but I notice that in my local city, the place is awash in taxis - I think the numbers would have to be regulated better and people encouraged to use buses/trams/trains/bikes instead. Ideally, the public transport would be (somehow.. I admit) set up so that the need for individual point to point transport would only rarely be required...
From my visits to Paris, I'd say the biggest detriment to driving there is other drivers. It's common to see parked cars completely blocked in (cars in front and behind leaving literally a centimeter of space between bumpers), and an obscenely high percentage of cars have all sorts of scuffs and bumps (presumably from having to force their way out of parking spaces when blocked in). I'm no car nut, but I wouldn't dream of taking my own car there.
I think the issue is the people, not the cars. The people of Peachtree City, GA have a large golf cart culture. And, they still have golf cart traffic jams[1], golf cart drunk drivers[2], and severe golf cart accidents [3].
They don't 'primarily' use golf carts. They still use mostly cars. I lived there for several months. Golf carts are definitely part of the culture, the city encourages them by building paths, but the vast majority still drive cars.