I lived in Prague, whose center is medieval and the neighbourhoods around it pre-1900, and even though what you say is true (fewer people drove everywhere), the streets were still saturated to their capacity.

It seemed to me that regardless of the city, many people will drive until the point where traffic jams and parking become a nightmare, and only then consider the alternatives. This point of pain is much lower in old European cities that weren't built as car-centric and much higher in the US, but the pattern seems to repeat itself.