I've long had the notion that we shouldn't drive anything heavier than a golf cart in cities. It would be a lot safer for everyone and even teenagers could drive them.
For some areas of the US (notably the southwest), the summer months are brutal. Air conditioning is virtually a requirement, not a luxury. Unless we mandate that all businesses install showers for employees (and we change as a society and not mind people being sweaty and smelly otherwise).
The majority of power used in a car during the summer months - beyond moving itself and passengers - comes from the AC system. These systems can sometimes exceed the BTU output of a house AC unit, mainly because they are fighting the heat input from the outside constantly, due to the lack of insulation in an automobile passenger cabin. So they have to be more powerful to offset that. Which means they need a power system to keep up, and historically that has been an IC engine. I'm not sure how an electric vehicle's AC system fares during a hot summer day, or how it effects the battery charge, but I'd expect it to be fairly heavy on that front.
Could we do as you suggest? Certainly we could; we did something similar (more or less) for hundreds of years before automotive AC was a standard item (in fact, the automobile was around a long time before AC became a readily available option). At the same time, though, certain social standards were different, plus there's the argument that cities weren't as built up or as dense (heat island effect), making them (marginally) cooler than today.
Here in Phoenix, though - it starts to become a furnace around the end of May, and continues until about the end of October. There are nights where it never falls below 100F at times; most of the time it never falls below 90F at night. There isn't an easy solution for comfortable personal transportation in such conditions, except for the automobile as we currently know it.