I think you are forgetting about the fact that we can store energy. This will act as a buffer and not run into the problems you are describing. Of course the buffer size needs to be reasonable and we need to collect and store energy at a rate greater than or equal to the rate of consumption, but I think we can achieve now with current technology.
Energy storage solutions are getting better, but energy storage research didn't really pickup till the 1990s.
Pumped Hydro (~80% efficiency) is still the best large-scale energy storage mechanism in the USA. Maybe in a few years Redox-Flow Batteries (Or flywheels, or Lithium, or Compressed Air, or...) will catch up to technologies from the 1960s and we can have advanced energy storage.
But for now, the vast vast majority of energy storage is basically pumping water up a hill and letting it back down at a later time. Something like 95% of the entire USA's energy storage is pumped hydro.
