I am often struck by the way that the current debate about intermittent renewable power is strikingly similar to the arguments between net heads and bell heads two decades previously. The funny part from a historical standpoint is that both were kind of right.
You also miss an important point. The other factor that is important in power generation is if it is dispatchable. By dispatchable I mean can she adjust the power generated quickly to meet demand. Current nuclear and coal plants require long startup times and current nuclear plants can't throttle their power output very well, which makes them much less valuable in a world with a lot of renewables. Combined-cycle natural gas, on the other hand, is easy and quick to start up so it is very dispatchable.
There are a few other factors that somewhat mitigate the intermittent nature of solar and wind. The first one, kind of obvious, is that you know more or less in the near future how much power you will be able to produce from these sources (we know when the sun rises and sets, and weather forecasts 24 hours out are fairly accurate -- especially if you just want to know if it will be sunny or windy). The other is that if we have a larger geographical distribution for solar and wind, the intermittency problem is somewhat mitigated -- it is unlikely to be cloudy and windless everywhere at the same time. Finally, there are other ways to store energy than batteries. If you have an old-style hot water heater rather than an on-demand system, you are essentially storing energy in the hot water tank -- and it would be plausible to have a system that would heat your hot water when the sun is shining or the wind is blowing. You can do similar things with heating and air conditioning systems in buildings, and even to a lesser extent in refrigerators or freezers.
You will still need some storage, but probably not as much as you think.