Most nations still have peak demand infrastructure setup for daytime industrial power demand, most of which already requires high capacity delivery. That demand hugely drops off in the evenings as factories shut down for the day and workers return home.
Most new EV power demand happens in the evenings, "filling in the bathtub" (because the average demand graph looks like a bathtub) that the industrial demand curve leaves behind. The difference between industrial (daytime/peak) and consumer (evening/off-peak) demands is a big enough bathtub that estimates are that it would be a while before EV cars get even close to filling in the bathtub.
One estimate I saw was that if every car on the road switched to an EV car tomorrow and charged only at off-peak times, it would almost fill the bathtub, and thus require no actual generation increase.
That doesn't deal with the infrastructure chicken-and-egg issues of street-level and parking structure access to car charging, of course.