You can have several connections at the same time, somehow the power plants might struggle to give a lot of power at one moment and barely any power the next moment.
Usually, problem is not with the power plants, it's the low voltage (last) transformer capacity. (for example from 10kV down to 0,4kV that SuperChargers eat). If charging sites have their own transformers, most of the problem is solved. 350kW at 10 000V is just 12A per phase. Piece of cake. Also overload is a rare event (in terms of seconds per day). Power companies are happy to share data at which particular minute NOT to have charging happening. For example UK with football commercial teatime seconds or anywhere where Sun eclipse happens. Dynamic load adjustment is the key. Some huge companies already listen for grid commands (for example refineries).