There's no "shifting" going on. This is just an additional power usage to their normal usage that has already existed.
Yes and no. In other words, your statement above shouldn't be taken in the absolute.
Power utilities have some plants that operate efficiently that are what they call "steady power", and other plants that provide "reserve power" that they turn on or crank up higher when needed. In Alabama, fighting the heat during the day, combined with power to offices and people being up in homes during the day and using more power is the main culprit power companies are having to satisfy. They don't want EV's added to that part of the day. At night there's often excess steady power on the grid not utilized while everyone is asleep.
So yes, EV's are an additional power usage and yes, us conservatives are right that the grid can't handle it. But from the utility engineers' perspective, their problem of adding a bunch of EV's to the grid demand is lessened some if most EV's are charging at night when on many nights there's already a lot of steady power not being utilized. Hence the incentive of a lower rate if charging at night.