Remember, the internal combustion engine is terribly inefficient. It was attractive because it's small enough to fit in a personal vehicle and the liquid fuel has so much energy that you can throw most of it away without bankrupting yourself, but that means there's a big difference between how much energy is used by a conventional ICE car and how much electricity is needed (and thus passes through the grid) for an electric car with the same power.

You might easily get rid of 10GW of ICE cars and only need to buy 2GW of electrical grid capacity to replace that.

There are lots of more immediately noticeable consequences while you wait decades for any global climate effects, the air on city streets is better, a bunch of land used to store incredibly dangerous liquid fuel where people want to buy it is released for new housing, maintenance costs on cars fall. I don't think it'll take long for the story to be "Why didn't we do this sooner?" rather than "Why did we bother trying this?"