“everyone can plug in at the same time, the cars will simply charge at the rate the grid can handle.”
No, that’s not the way electrical grids work.
Not the grids as a whole, but EV charging systems can adapt the charge rate, and thus the grid load, to what's available. This would require coordination between those systems, the other stuff that wants some of those electrons, and the substation feeding the area.
A lot of people seem to think that if this Mercedes is able to charge at 350kW, we'd need that capacity at every domestic charge socket too. Which is utterly daft. You want to charge a car at maximum power when 'filling up' at an actual roadside service station. At a domestic connection there's usually more time and less current available to charge, and the charge controller (which is in the car) can, and should, adapt to that.