The challenges of supporting a nation of EV is entirely different than gasoline/diesel. Storage of electricity isn't as simple as burrying a large tank, and when everyone stops for gas at the same time it has no impact on the electric grid.
We already have growing pains in our electric grid, those would only get worse with surging use as vehicles charge. If the existing grid may not work and storage is impractical, what's left?
Many charging stations already have batteries on site, because it lets them use cheaper electricity and/or provide charging speeds faster than their grid connection.
EVs used for commuting are very flexible with their charging times. They can automatically delay/schedule charge times. Median US commute (35mi) requires charging about once a week. People who can't charge at home could charge at work (such low-speed chargers are relatively cheap and easy to install).
I would be surprised if the grid + new battery storage could handle so much peak-y demand being added to it, but I've been surprised before!